research article a note on primitivity of ideals in skew...
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Research ArticleA Note on Primitivity of Ideals in Skew PolynomialRings of Automorphism Type
Edilson Soares Miranda
Departamento de Ciencias Centro de Ciencias Exatas Universidade Estadual de Maringa 87360-000 Goioere PR Brazil
Correspondence should be addressed to Edilson Soares Miranda esmirandauembr
Received 4 March 2016 Accepted 11 May 2016
Academic Editor Kaiming Zhao
Copyright copy 2016 Edilson Soares Miranda This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We extend results about primitive ideals in polynomial rings over nil rings originally proved by Smoktunowicz (2005) for 120590-primitive ideals in skew polynomial rings of automorphism type
1 Introduction
Throughout this paper 119877 denotes an associative ring but doesnot necessarily have an identity element and 120590 119877 rarr 119877
an automorphism of 119877 unless otherwise stated We denoteby 119877[119909 120590] the skew polynomial rings of automorphism typewhose elements are polynomials sum
119899
119894=0119886119894119909119894 119886119894isin 119877 for every
119894 ge 0 with usual addition and the following multiplication119909119886 = 120590(119886)119909 for all 119886 isin 119877
A ring 119877 is said to be a Jacobson ring if every primeideal of 119877 is an intersection of (either left or right) primitiveideals of 119877 In [1] Smoktunowicz proved that if 119877 is a nilring and 119868 an ideal of 119877[119909] then 119877[119909]119868 is Jacobson radicalif and only if 119877[119909]119868
1015840
[119909] is Jacobson radical where 1198681015840 is the
ideal of 119877 generated by coefficients of polynomial from 119868Also if 119877 is a nil ring and 119868 is a primitive ideal of 119877[119909] then119868 = 119872[119909] for some ideal 119872 of 119877 and affirmative answer tothis question is equivalent to the Kothe conjecture Our mainresults state that if119877 is a nil ring and 119868 an ideal of119877[119909 120590] then119877[119909 120590]119868 is 120590-Jacobson radical if and only if 119877[119909 120590]119868
1015840
[119909 120590]
is 120590-Jacobson radical where 1198681015840 is the ideal of 119877 generated by
coefficients of polynomial from 119868 Also if 119877 is a nil ring and119868 is a 120590-primitive ideal of 119877[119909 120590] then 119868 = 119872[119909 120590] for someideal 119872 of 119877 This result includes as particular cases all theabove results
Nowwe recall some terminology and results see [2ndash4] Aright ideal 119876 of a ring 119877 is called modular in 119877 if and only ifthere exists an element 119887 isin 119877 such that 119886 minus 119887119886 isin 119876 for every119886 isin 119877 An ideal 119868 of a ring 119877 is said to be a 120590-invariant if
and only if 120590(119868) = 119868 An ideal 119875 of 119877 is said to be a right 120590-primitive in 119877 if and only if there exists a modular maximalright ideal 120590-invariant119876 of119877 such that119875 is themaximal idealcontained in 119876 For 119891 isin 119877[119909 120590] deg(119891) denotes the degreeof 119891 and lc(119891) the leading coefficient of 119891
2 Results
We begin with the following results that extend ([1Lemma 1]) and the proof is also similar to the one in thepaper
Lemma 1 Let 119877 be a ring 119869 a right ideal of 119877 119891 isin 119869[119909 120590] 119876a right ideal of 119877[119909 120590] and 119887 isin 119877[119909 120590] such that 119886 minus 119887119886 isin 119876
for every 119886 isin 119877[119909 120590] If 119887 minus 119891119909 isin 119876 then for every 119894 ge 1 thereare 119891119894isin 119869[119909 120590] such that 119887 minus 119891
119894119909119894
isin 119876 and deg(119891119894) le deg(119891)
Proof We proceed by induction on 119899 If 119899 = 1 we put1198911= 119891
Suppose the lemma holds for some 119899 ge 1 Let
119891119899= 1198860+ 1198861119909 + sdot sdot sdot + 119886
119896119909119896
isin 119869 [119909 120590] (1)
with 119887 minus 119891119899119909119899
isin 119876 and 119896 le deg(119891) Consider
119891119899+1
= 119891120590 (1198860) + 1198861+ 1198862119909 + sdot sdot sdot + 119886
119896119909119896minus1
isin 119869 [119909 120590] (2)
Since 119887 minus 119891119909 isin 119876 then 119891119909 = 119887 + 119902 119902 isin 119876 Thus
119887 minus 119891119899+1
119909119899+1
= 119887 minus 119891119899119909119899
+ (1198860minus 1198871198860) 119909119899
minus 1199021198860119909119899
isin 119876 (3)
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2016 Article ID 7159180 5 pageshttpdxdoiorg10115520167159180
2 International Journal of Mathematics and Mathematical Sciences
We denote by 1198771 the usual extension of 119877 to a ring with
identity and by 120590 again the natural extension of 120590 to 1198771
The next lemma extends ([1 Lemma 2])
Lemma 2 Let 119868 be an ideal of 119877[119909 120590] with 120590(119868) = 119868 and 119869 aright ideal of 119877 with 120590(119869) = 119869 Consider 119901 = 119886
0+ 1198861119909 + sdot sdot sdot +
119886119896119909119896
isin 119868 119896 gt 0 and
119880 = sum
119894isinZ119869 [119909 120590] 120590
119894
(119886119896) 1198771
[119909 120590] (4)
(i) If ℎ isin 119880119897 119897 ge 1 and deg(ℎ) ge 119896 then there exists
119892 isin 119880119897minus1 such that ℎ minus 119892 isin 119868 and deg(119892) lt deg(ℎ)
(ii) Let 119876 be a right ideal of 119877[119909 120590] 119887 isin 119877[119909 120590] such that119886minus119887119886 isin 119876 for every 119886 isin 119877[119909 120590] and 119868 sub 119876 If 119887minus119891119909 isin
119876 with 119891 isin 119869[119909 120590] deg(119891) ge 1 and 119887 minus 119892 isin 119876 where119892 isin 119880
deg(119891) then for every 119894 gt deg(119892) there exists119892119894isin 119869[119909 120590] such that 119887 minus 119892
119894119909119894
isin 119876 and deg(119892119894) lt 119896
Proof (i) Let ℎ = 1198880+ 1198881119909 + sdot sdot sdot + 119888
119905119909119905
isin 119880119897 119888119905
= 0 and 119896 le 119905We can write
119888119905=
1198981015840
sum
119895=0
120572119895120573119895 120572119895isin 119880 120573
119895isin 119880119897minus1
(5)
Then
120572119895=
119899119895
sum
119894=0
119905119895119894120590119902119895119894(119886119896) 119906119895119894 119905119895119894
isin 119869 119906119895119894
isin 1198771
119902119894119895isin Z (6)
Hence
119888119905=
1198981015840
sum
119895=0
119899119895
sum
119894=0
(119905119895119894120590119902119895119894(119886119896) 119906119895119894120573119895) =
119898
sum
119894=0
119901119894120590119897119894(119886119896) 119890119894119902119894
(7)
with 119901119894isin 119869 119890119894 119902119894isin 1198771 119902119894isin 119880119897minus1 and 119897
119894isin Z Put
119892 = ℎ minus 119888119905119909119905
+
119898
sum
119894=0
119901119894(120590119897119894(119901) minus 120590
119897119894(119886119896) 119909119896
) 120590minus119896
(119890119894119902119894) 119909119905minus119896
(8)
Therefore
119892 minus ℎ =
119898
sum
119894=0
119901119894(120590119897119894(119901)) 120590
minus119896
(119890119894) 120590minus119896
(119902119894) 119909119905minus119896
isin 119868 (9)
Since 119869[119909 120590]119880119897minus1
sub 119880119897minus1 and ℎ isin 119880
119897minus1 then 119892 isin 119880119897minus1 and
deg(119892) lt deg(ℎ)(ii) By Lemma 1 for every 119894 ge 1 there exists 119891
119894isin 119869[119909 120590]
such that
119887 minus 119891119894119909119894
isin 119876 deg (119891119894) le deg (119891) (10)
Consider
119892 =
119898
sum
119895=0
119888119895119909119895
119888119895isin 119880
deg(119891) (11)
For every 119899 gt 119898 denote
ℎ119899=
119898
sum
119895=0
119891119899minus119895
120590119899minus119895
(119888119895) isin 119869 [119909 120590] cap 119880
deg(119891) (12)
Note that deg(ℎ119899) le deg(119891
119899minus119895) le deg(119891) thus for every 119894 ge 1
119891119894119909119894
= 119887 + 119902119894 119902119894isin 119876 (13)
Hence
119887 minus ℎ119899119909119899
= 119887 minus 119887
119898
sum
119895=0
119888119895119909119895
+
119898
sum
119895=0
119902119899minus119895
119888119895119909119895
isin 119876 (14)
Because 119887 minus 119892 isin 119876 and 119892 minus 119887119892 isin 119876 then 119887 minus 119887119892 isin 119876 We havethat for every 119899 gt 119905 there exists ℎ
119899isin 119880
deg(119891)sube 119869[119909 120590] such
that 119887 minus ℎ119899119909119899
isin 119876 If deg(ℎ119899) lt 119896 then ℎ
119899is the 119892
119899required
If deg(ℎ119899) ge 119896 by first part of this lemma there exists
1205821198991
isin 119880deg(119891)minus1
sube 119869 [119909 120590] (15)
such that ℎ119899minus1205821198991
isin 119868 anddeg(1205821198991
) lt deg(ℎ119899)Thus 119887minus120582
1198991
119909119899
isin
119876 for all 119899 gt 119898 If deg(1205821198991
) lt 119896 then 1205821198991
is the 119892119899required If
deg(1205821198991
) ge 119896 using similar arguments as above we can find119904 isin N such that
120582119899119904
isin 119880deg(119891)minus119904
sube 119869 [119909 120590] (16)
with 119887 minus 120582119899119904
119909119899
isin 119876 and deg(120582119899119904
) le deg(119891) minus 119904 lt 119896 for every119899 gt 119898 Hence 120582
119899119904
is the 119892119899required
Let 119903 isin 119877 119876 a right ideal of 119877[119909 120590] 120590(119876) = 119876 and 119887 isin
119877[119909 120590] such that 119886 minus 119887119886 isin 119876 for all 119886 isin 119877[119909 120590] Following [1]we have the followingWe say that V is a ldquogood number for 119903rdquoif for all sufficiently large 119899 there are 119891
119899isin 119877[119909 120590] such that
119887 minus 119903119891119899119909119899
isin 119876 with deg(119891119899) le V Let 119860 sube 119877 we denote
= 119886 isin 119860 minus 119876 | 120590 (119886) minus 119886 isin 119876 (17)
Lemma 3 Let 119876 be a right ideal of 119877[119909 120590] maximal in theset of all right ideals 120590-invariants with 119887 isin 119877[119909 120590] such that119886 minus 119887119886 isin 119876 for all 119886 isin 119877[119909 120590] Suppose 119891 isin 119877[119909 120590] with119887minus119891119909
119895
isin 119876 for some 119895 ge 1 If there is no right ideal 119869 of 119877 with120590(119869) = 119869 119869 sube 119876 and 119869 = 119877 then there exists a positive integerV and 119903 isin such that if119908 isin 119903119877[119909 120590]with 119887minus119908119909
119898
isin 119876119898 ge 0and deg(119908) le V then 119897119888(119908) isin 119903119877 119897119888(119908)(119876 cap 119877) sube 119876 and V isa good number for all 119886 isin 119903119877
Proof Let V be minimal positive integer such that there exists1199081015840
= 1198940+1198941119909+sdot sdot sdot+119903119909
Visin 119877[119909 120590] and119898 ge 1with 119887minus119908
1015840
119909119898
isin 119876
and deg(1199081015840) = V It is clear that 119903 notin 119876 If 119888 = 120590(119903) minus 119903 notin 119876 put119892 = 120590(119908
1015840
) minus 1199081015840 and
119860 = sum
119894isinZ120590119894
(119888) 1198771
(18)
Thus 119860 is a right ideal of 119877 with 120590(119860) = 119860 and 119860 sube 119876 Byassumption 119860 = 119877 then 119903 = sum
119904
119894=0120590119902119894(119888)119897119894 where 119897
119894isin 1198771 and
119902119894isin Z Put
119905 = 1199081015840
minus
119904
sum
119894=0
120590119902119894(119892) 120590minusV
(119897119894) (19)
International Journal of Mathematics and Mathematical Sciences 3
Comparing the leading coefficients of 1199081015840 and
sum119904
119894=0120590119902119894(119892)120590minusV(119897119894) we have that
119887 minus 119905119909119898
isin 119876 deg (119905) le V minus 1 (20)
which contradicts theminimality of VTherefore 120590(119903)minus119903 isin 119876consequently 119903 isin
Suppose that 119903119902 notin Q for some 119902 isin 119877 cap 119876 Put 1198921015840
=
1199081015840
120590minusV(119902) isin 119876 using similar arguments as above we can have
a contradiction Hence 119903(119876 cap 119877) sube 119876If there exists 119908 isin 119903119877[119909 120590] with 119887 minus 119908119909
119895
isin 119876 119895 ge 0 anddeg(119908) le V then using similar arguments as above we canshow that lc(119908) isin 119903119877 and lc(119908)(119876 cap 119877) ⫅ 119876 Moreover if119886 isin 119903119877 put 119861 = 119886119877 + 119876 cap 119877 we have that 119861 is a right ideal of119877 with 120590(119861) = 119861 and 119861 sube 119876
By assumption 119861 = 119886119877 + 119876 cap 119877 = 119877 Thus 1199081015840
= 11988611990810158401015840
+
1199021015840 where 119908
10158401015840
isin 119877[119909 120590] deg(11990810158401015840) le deg(1199081015840) and 1199021015840
isin 119876Therefore 119887 minus 119886119908
10158401015840
119909119898
isin 119876 Consequently V is a good numberfor all 119886 isin 119903119877
Lemma 4 Let 119869 be a right ideal 119877 with 120590(119869) = 119869 119869 sube 119876 and119869 = 119877 such that for all sufficiently large 119899 there are 119891
119899isin 119869[119909 120590]
such that 119887 minus 119891119899119909119899
isin 119876 and deg(119891119899) le 119896 where 119876 is a right
ideal of 119877[119909 120590] and 119887 isin 119877[119909 120590] such that 119886 minus 119887119886 isin 119876 for every119886 isin 119877[119909 120590] Then there exists a positive integer V and 119903 isin
such that if 119908 isin 119903119877[119909 120590] with 119887 minus 119908119909119898
isin 119876 119898 ge 0 anddeg(119908) le V one has that 119897119888(119908) isin 119903119877 119897119888(119908)(119876 cap 119877) sube 119876 and Vis a good number for all 119886 isin 119903119877
Proof Let V be minimal positive integer such that for allsufficiently large119899 there are119891
119899isin 119869[119909 120590] such that 119887minus119891
119899119909119899
isin 119876
and deg(119891119899) le V Put
1199081015840
= 1198940+ 1198941119909 + sdot sdot sdot + 119894Vminus1119909
Vminus1+ 119903119909
Visin 119869 [119909 120590] (21)
with 119887 minus1199081015840
119909119898
isin 119876119898 ge 0 and deg(1199081015840) le V By Lemma 1 andminimality of V we have that 119903 notin 119876 Using the same ideas ofLemma 3 we have that 119903 isin and 119903(119876cap119877) sube 119876 Since 119903119877 sube 119869we have that the first part of lemma is satisfied
Let 119886 isin 119903119877 sube we denote by 119861 the right ideal of 119877
119861 = sum
119894isinZ
120590119894
(119886) 1198771
120590 (119861) = 119861 119861 sube 119869 119861 sube 119876 (22)
For sufficiently large 119899 there are 119892119899isin 119861[119909 120590] sube 119869[119909 120590] such
that 119887 minus 119892119899119909119899
isin 119876 and deg(119892119899) le V Put
119892119899= 1198881198990
+ 1198881198991
119909 + sdot sdot sdot + 119888119899V119909Visin 119861 [119909 120590] (23)
For every 0 le 119895 le V we have that 119888119899119895
= sum
119898119895
119894=0120590119902119899119894 (119886)119897119899119894
where119897119899119894
isin 1198771 and 119902
119899119894
isin Z Consequently
119888119899119895
=
119898119895
sum
119894=0
(120590119902119899119894 (119886) minus 119886) 119897
119899119894
+ 119886
119898119895
sum
119894=0
119897119899119894
(24)
Since 119886 isin 119903119877 we can write
119888119899119895
= 119904119899119895
+ 119903119899119895
119904119899119895
isin 119876 cap 119877 119903119899119895
isin 119877 (25)
Put ℎ119899= 1199031198990
+1199031198991
119909+ sdot sdot sdot + 119903119899V119909V thus 119887minus119886ℎ
119899119909119899
isin 119876 ThereforeV is a good number for all 119886 isin 119903119877
Lemma 5 Let 119876 be a right ideal of 119877[119909 120590] 119887 isin 119877[119909 120590] suchthat 119886minus 119887119886 isin 119876 for all 119886 isin 119877[119909 120590] and V is good number for all119886 isin 119903119877 where 119903 isin Assume that for every 119908 isin 119903119877[119909 120590] with119887 minus 119908119909
119898
isin 119876 119898 ge 0 and deg(119908) le V one has that 119897119888(119908) isin 119903119877
and 119897119888(119908)(119876 cap 119877) sube 119876 If there are 119901 and 1199011015840
isin 119903119877 with
(119901119877 + 119876 cap 119903119877) cap (1199011015840
119877 + 119876 cap 119903119877) sube 119876 (26)
then V minus 1 is a good number for 119903
Proof Since V is a good number for 119901 and 1199011015840 then for every
sufficiently large 119899 there are 119892119899isin 119901119877[119909 120590] and 119892
1015840
119899isin 1199011015840
119877[119909 120590]
such that
119887 minus 119892119899119909119899
isin 119876 (27)
119887 minus 1198921015840
119899119909119899
isin 119876 (28)
with deg(119892119899) deg(1198921015840
119899) le V Consider
119892119899= 1199011198990
+ 1199011198991
119909 + sdot sdot sdot + 119901119899V119909V 119901119899V
isin 119901119877
1198921015840
119899= 1199011015840
1198990
+ 1199011015840
1198991
119909 + sdot sdot sdot + 1199011015840
119899V119909V 1199011015840
119899Visin
1199011015840
119877
(29)
Since 119901119899V
minus 1199011015840
119899Visin 119876 then
119901119899V
isin (119901119877 + 119876 cap 119903119877) cap (1199011015840
119877 + 119876 cap 119903119877) sube 119876 (30)
a contradictionThus there exists sufficiently large 119894 isin N such that 119888 =
119901119894Vminus 1199011015840
119894Visin 119903119877 hence V is a good number for 119888 Then for all
sufficiently large 119899 there are ℎ119899isin 119877[119909 120590] such that 119887minus119888ℎ
119899119909119899
isin
119876 and deg(ℎ119899) le V We denote
ℎ119899= 1199031198990
+ 1199031198991
119909 + sdot sdot sdot + 119903119899V119909V (31)
Consider
119896119899= 119888ℎ119899+ (1198921015840
119894minus 119892119894) 120590minusV
(119903119899V) isin 119903119877 [119909 120590] (32)
Since 1198921015840
119894minus 119892119894isin 119876 then 119887 minus 119896
119899119909119899
isin 119876 Moreover
119896119899= 119888ℎ119899minus 119888119903119899V119909V+
Vminus1
sum
119895=0
(1199011015840
119894119895
minus 119901119894119895
) 119909119895
120590minusV
(119903119899V) (33)
Consequently V minus 1 is a good number for 119903
The following theorem extends ([1 Theorem 1])
Theorem 6 Let 119877 be a nil ring and let 119868 be a 120590-primitive idealin 119877[119909 120590] Then 119868 = 119868
1015840
[119909 120590] where 1198681015840 is an ideal 120590-invariant
of 119877
Proof Assume by contradiction that there are 1198860 1198861 119886
119896isin
119877 with
1198860+ 1198861119909 + sdot sdot sdot + 119886
119896119909119896
isin 119868 119886119896notin 119868 (34)
4 International Journal of Mathematics and Mathematical Sciences
Since 119868 is a 120590-primitive ideal in119877[119909 120590] there is a right ideal119876of119877[119909 120590]with 120590(119876) = 119876 and 119887 isin 119877[119909 120590] such that 119886minus119887119886 isin 119876
for all 119886 isin 119877[119909 120590] Moreover 119876 is a maximal in the set ofright ideals 120590-invariants and 119868 is the maximal ideal containedin 119876 We have that 119877[119909 120590]119909⫅119876 otherwise 119887 isin 119877 which isimpossible because119877 is a nil ring By definition of119876 it followsthat 119877[119909 120590]119909 + 119876 = 119877[119909 120590]
If 119887 minus ℎ119909119894
isin 119876 for some 119894 ge 0 with ℎ isin 119877[119909 120590] thendeg(ℎ) ge 1 In fact if ℎ isin 119877 let 119905 ge 1 be the minimal positiveinteger with respect to ℎ
119905
isin 119876 Thus (119887 minus ℎ119909119894
)120590minus119894
(ℎ119905minus1
) isin 119876Then 119887120590
minus119894
(ℎ119905minus1
) isin 119876 hence 120590minus119894
(ℎ119905minus1
) isin 119876 Consequentlyℎ119905minus1
isin 119876 a contradictionLet 119869 be a right ideal of 119877 with 120590(119869) = 119869 and 119869 sube 119876 We
have that 119869[119909 120590]119909+119876 = 119877[119909 120590]There exists119891 isin 119869[119909 120590] suchthat 119887 minus 119891119909 isin 119876 Consider
119880 = sum
119894isinZ
119869 [119909 120590] 120590119894
(119886119896) 1198771
[119909 120590] (35)
Since 119868 is an ideal 120590-prime and 119886119896
notin 119868 then 119880 sube 119868Consequently 119880 sube 119876 because 119868 is the maximal idealcontained in 119876 Then 119880
deg(119891)+ 119876 = 119877[119909 120590] There exists
1198921015840
isin 119880deg(119891) such that 119887 minus 119892
1015840
isin 119876 By Lemma 2 for every119894 ge deg(1198921015840) there are 119892
1015840
119894isin 119869[119909 120590] such that 119887 minus 119892
1015840
119894119909119894
isin 119876 anddeg(1198921015840
119894) lt 119896 Lemmas 3 and 4 imply that there are 119903
1015840
isin andV1015840 ge 1 such that if119908 isin 119903
1015840
119877[119909 120590]with 119887minus119908119909119898
isin 119876119898 ge 1 anddeg(119908) le V1015840 then lc(119908) isin
1199031015840
119877 and lc119908(119876cap119877) sube 119876 MoreoverV1015840 is a good number for all 119886 isin
1199031015840
119877 Let V beminimal such thatV is a good number for all 119886 isin
1199031015840
119877 We have that V le V1015840 Let119903 isin
1199031015840
119877 Since V is a good number for 119903 then for sufficientlylarge 119899 there are ℎ
119899isin 119877[119909 120590] such that
119887 minus 119903ℎ119899119909119899
isin 119876 deg (ℎ119899) le V (36)
Consider 119891119899= 119903ℎ119899 then 119887 minus 119891
119899119909119899
isin 119876 and deg(119891119899) le V For
some 119894 isin N there are 119891119894 119891119894+1
119891119894+119896
isin 119903119877[119909 120590] such that119887 minus 119891119895119909119895
isin 119876 deg(119891119895) le V and 119894 le 119895 le 119894 + 119896 Put
119891119895= 1199031198861198950
+ 1199031198861198951
119909 + sdot sdot sdot + 119903119886119895V119909V= 119892119895+ 119888119895119909V (37)
where 119892119895
= 1199031198861198950
+ 1199031198861198951
119909 + sdot sdot sdot + 119903119886119895Vminus1
119909Vminus1
isin 119903119877[119909 120590] and119888119895= 119903119886119895V Since deg(119891
119895) le V le V1015840 then 119888
119895notin 119876 Moreover
120590 (119888119895) minus 119888119895isin 119876 119888
119895(119876 cap 119877) sube 119876 (38)
Since 119877 is a nil ring consider 119890119895= 119888
119899119895
119895 where 119899
119895is a minimal
with respect to the condition 119888
119899119895
119895notin 119876 Thus 120590(119890
119895) minus 119890119895isin 119876 for
all 119894 ge 0 We have that
119891119895120590119895
(119890119895) = 119892119895120590119895
(119890119895) + 119888119895120590119895+V
(119890119895) 119909
V
= 119892119895120590119895
(119890119895) + 119888119895(120590119895+V
(119890119895) minus 119890119895) 119909
V
+ 119888119895119890119895119909V
(39)
Put 119905119895= 119892119895120590119895
(119890119895) isin 119903119877[119909 120590] Thus
119891119895120590119895
(119890119895) minus 119905119895isin 119876 deg (119905
119895) le V minus 1 (40)
for every 119894 le 119895 le 119894 + 119896 Since 119890119895isin 119903119877 sube
1199031015840
119877 if V minus 1 is not agood number for 119903 then Lemma 5 implies that
119894+119896
⋂
119895=1
(119890119895119877 + 119876 cap 119903119877) ⫅119876 (41)
In this case there exists 119904 isin ⋂119894+119896
119895=1(119890119895119877+119876cap119903119877) such that 119904 notin 119876
Consequently 119904minus 119890119895119889119895isin 119876cap119903119877 119889
119895isin 119877 and 119890
119895119889119895isin 119890119895119877 Then
119904 isin 119903119877 sube1199031015840
119877 Therefore V is a good number for 119904 Then forsufficiently large 119899 there are 119891
119899isin 119904119877[119909 120590] such that
119887 minus 119891119899119909119899
isin 119876 deg (119891119899) le V (42)
Let
119891119899=
V
sum
119895=0
119904119887119895119899
119909119895
(43)
Since 119887 minus 119891119895119909119895
isin 119876 119904 minus 119890119895119889119895
isin 119876 and 119890119895119889119895minus 119887119890119895119889119895
isin 119876then (119887 minus 119891
119895119909119895
)119890119895119889119895isin 119876 Thus 119887119890
119895119889119895minus 119891119895119909119895
119890119895119889119895isin 119876 hence
119904 minus 119891119895119909119895
119890119895119889119895isin 119876 for every 119894 le 119895 le 119894 + 119896
Let
119892119899=
V
sum
119895=0
119891119894+Vminus119895120590
119894+Vminus119895(119890119894+Vminus119895119889119894+Vminus119895119887119895
119899
) 119909119894+V
isin 119903119877 [119909 120590] (44)
We have that 119891119899minus119892119899isin 119876 Thus 119887minus119892
119899119909119899
= (119887minus119891119899119909119899
) + (119891119899minus
119892119899)119909119899
isin 119876 Put
ℎ119899=
V
sum
119895=0
119905119894+Vminus119895120590
119894+Vminus119895(119889119894+Vminus119895119887119895
119899
) isin 119903119877 [119909 120590] (45)
We can write 119887 minus ℎ119899119909119894+V+119899 as
119887 minus
V
sum
119895=0
(119905119894+Vminus119895 minus 119891
119894+Vminus119895120590119894+Vminus119895
(119890119894+Vminus119895)) 120590
119894+Vminus119895(119889119894+Vminus119895119887119895
119899
)
sdot 119909119894+V+119899
minus 119892119899119909119899
(46)
Thus for all sufficient large 119899
119887 minus ℎ119899119909119894+V+119899
isin 119876 deg (ℎ119899) le V minus 1 (47)
Then V minus 1 is a good number for all 119903 isin1199031015840
119877 This contradictsthe minimality of V
Recall that the 120590-Jacobson radical 119869120590(119877) of a ring 119877 is
defined as the intersection of all 120590-primitive ideals of 119877 Aring 119877 is a 120590-Jacobson radical if 119869
120590(119877) = 119877
Theorem 7 Let 119877 be a nil ring and let 119868 be an ideal of 119877[119909 120590]Consider 119868 the ideal of119877 generated by coefficients of polynomialfrom 119868 Then 119877[119909 120590]119868[119909 120590] is 120590-Jacobson radical if and onlyif 119877[119909 120590]119868 is 120590-Jacobson radical
International Journal of Mathematics and Mathematical Sciences 5
Proof Assume by contradiction that 119877[119909 120590]119868 is not 120590-Jacobson radical Then there is a 120590-primitive ideal 119875 of119877[119909 120590]119868 such that 119875 = 119877[119909 120590]119868 We have that there is anideal 119870 of 119877[119909 120590] such that 119875 = 119870119868 Therefore 119870 is a 120590-primitive ideal of 119877[119909 120590] By Theorem 6 there is an ideal 119875of 119877 such that119870 = 119875[119909 120590] It is clear that 119868 sube 119875 Since
(119877 [119909 120590] 119868 [119909 120590])
(119875 [119909 120590] 119868 [119909 120590])
≃
119877 [119909 120590]
119870
(48)
then 119875[119909 120590]119868[119909 120590] is a 120590-primitive ideal a contradictionUsing the fact that 119868 sube 119868[119909 120590] the converse follows
Corollary 8 If 119877 is a nil ring then the polynomial ring of typeautomorphism 119877[119909 120590] can not be homomorphically mappedonto a 120590-simple 120590-primitive ring
Competing Interests
The author declares that they have no competing interests
References
[1] A Smoktunowicz ldquoOn primitive ideals in polynomial ringsover nil ringsrdquo Algebras and Representation Theory vol 8 no1 pp 69ndash73 2005
[2] E Cisneros M Ferrero and M I Conzles ldquoPrime ideals ofskew polynomial rings and skew laurent polynomial ringsrdquoMathematical Journal of Okayama University vol 32 pp 61ndash721990
[3] NDivinskyRings and Radicals Allen andUnwin London UK1965
[4] T Y Lam A First Course in Noncommutative Rings GraduateTexts in Mathematics Springer New York NY USA 1991
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Stochastic AnalysisInternational Journal of
2 International Journal of Mathematics and Mathematical Sciences
We denote by 1198771 the usual extension of 119877 to a ring with
identity and by 120590 again the natural extension of 120590 to 1198771
The next lemma extends ([1 Lemma 2])
Lemma 2 Let 119868 be an ideal of 119877[119909 120590] with 120590(119868) = 119868 and 119869 aright ideal of 119877 with 120590(119869) = 119869 Consider 119901 = 119886
0+ 1198861119909 + sdot sdot sdot +
119886119896119909119896
isin 119868 119896 gt 0 and
119880 = sum
119894isinZ119869 [119909 120590] 120590
119894
(119886119896) 1198771
[119909 120590] (4)
(i) If ℎ isin 119880119897 119897 ge 1 and deg(ℎ) ge 119896 then there exists
119892 isin 119880119897minus1 such that ℎ minus 119892 isin 119868 and deg(119892) lt deg(ℎ)
(ii) Let 119876 be a right ideal of 119877[119909 120590] 119887 isin 119877[119909 120590] such that119886minus119887119886 isin 119876 for every 119886 isin 119877[119909 120590] and 119868 sub 119876 If 119887minus119891119909 isin
119876 with 119891 isin 119869[119909 120590] deg(119891) ge 1 and 119887 minus 119892 isin 119876 where119892 isin 119880
deg(119891) then for every 119894 gt deg(119892) there exists119892119894isin 119869[119909 120590] such that 119887 minus 119892
119894119909119894
isin 119876 and deg(119892119894) lt 119896
Proof (i) Let ℎ = 1198880+ 1198881119909 + sdot sdot sdot + 119888
119905119909119905
isin 119880119897 119888119905
= 0 and 119896 le 119905We can write
119888119905=
1198981015840
sum
119895=0
120572119895120573119895 120572119895isin 119880 120573
119895isin 119880119897minus1
(5)
Then
120572119895=
119899119895
sum
119894=0
119905119895119894120590119902119895119894(119886119896) 119906119895119894 119905119895119894
isin 119869 119906119895119894
isin 1198771
119902119894119895isin Z (6)
Hence
119888119905=
1198981015840
sum
119895=0
119899119895
sum
119894=0
(119905119895119894120590119902119895119894(119886119896) 119906119895119894120573119895) =
119898
sum
119894=0
119901119894120590119897119894(119886119896) 119890119894119902119894
(7)
with 119901119894isin 119869 119890119894 119902119894isin 1198771 119902119894isin 119880119897minus1 and 119897
119894isin Z Put
119892 = ℎ minus 119888119905119909119905
+
119898
sum
119894=0
119901119894(120590119897119894(119901) minus 120590
119897119894(119886119896) 119909119896
) 120590minus119896
(119890119894119902119894) 119909119905minus119896
(8)
Therefore
119892 minus ℎ =
119898
sum
119894=0
119901119894(120590119897119894(119901)) 120590
minus119896
(119890119894) 120590minus119896
(119902119894) 119909119905minus119896
isin 119868 (9)
Since 119869[119909 120590]119880119897minus1
sub 119880119897minus1 and ℎ isin 119880
119897minus1 then 119892 isin 119880119897minus1 and
deg(119892) lt deg(ℎ)(ii) By Lemma 1 for every 119894 ge 1 there exists 119891
119894isin 119869[119909 120590]
such that
119887 minus 119891119894119909119894
isin 119876 deg (119891119894) le deg (119891) (10)
Consider
119892 =
119898
sum
119895=0
119888119895119909119895
119888119895isin 119880
deg(119891) (11)
For every 119899 gt 119898 denote
ℎ119899=
119898
sum
119895=0
119891119899minus119895
120590119899minus119895
(119888119895) isin 119869 [119909 120590] cap 119880
deg(119891) (12)
Note that deg(ℎ119899) le deg(119891
119899minus119895) le deg(119891) thus for every 119894 ge 1
119891119894119909119894
= 119887 + 119902119894 119902119894isin 119876 (13)
Hence
119887 minus ℎ119899119909119899
= 119887 minus 119887
119898
sum
119895=0
119888119895119909119895
+
119898
sum
119895=0
119902119899minus119895
119888119895119909119895
isin 119876 (14)
Because 119887 minus 119892 isin 119876 and 119892 minus 119887119892 isin 119876 then 119887 minus 119887119892 isin 119876 We havethat for every 119899 gt 119905 there exists ℎ
119899isin 119880
deg(119891)sube 119869[119909 120590] such
that 119887 minus ℎ119899119909119899
isin 119876 If deg(ℎ119899) lt 119896 then ℎ
119899is the 119892
119899required
If deg(ℎ119899) ge 119896 by first part of this lemma there exists
1205821198991
isin 119880deg(119891)minus1
sube 119869 [119909 120590] (15)
such that ℎ119899minus1205821198991
isin 119868 anddeg(1205821198991
) lt deg(ℎ119899)Thus 119887minus120582
1198991
119909119899
isin
119876 for all 119899 gt 119898 If deg(1205821198991
) lt 119896 then 1205821198991
is the 119892119899required If
deg(1205821198991
) ge 119896 using similar arguments as above we can find119904 isin N such that
120582119899119904
isin 119880deg(119891)minus119904
sube 119869 [119909 120590] (16)
with 119887 minus 120582119899119904
119909119899
isin 119876 and deg(120582119899119904
) le deg(119891) minus 119904 lt 119896 for every119899 gt 119898 Hence 120582
119899119904
is the 119892119899required
Let 119903 isin 119877 119876 a right ideal of 119877[119909 120590] 120590(119876) = 119876 and 119887 isin
119877[119909 120590] such that 119886 minus 119887119886 isin 119876 for all 119886 isin 119877[119909 120590] Following [1]we have the followingWe say that V is a ldquogood number for 119903rdquoif for all sufficiently large 119899 there are 119891
119899isin 119877[119909 120590] such that
119887 minus 119903119891119899119909119899
isin 119876 with deg(119891119899) le V Let 119860 sube 119877 we denote
= 119886 isin 119860 minus 119876 | 120590 (119886) minus 119886 isin 119876 (17)
Lemma 3 Let 119876 be a right ideal of 119877[119909 120590] maximal in theset of all right ideals 120590-invariants with 119887 isin 119877[119909 120590] such that119886 minus 119887119886 isin 119876 for all 119886 isin 119877[119909 120590] Suppose 119891 isin 119877[119909 120590] with119887minus119891119909
119895
isin 119876 for some 119895 ge 1 If there is no right ideal 119869 of 119877 with120590(119869) = 119869 119869 sube 119876 and 119869 = 119877 then there exists a positive integerV and 119903 isin such that if119908 isin 119903119877[119909 120590]with 119887minus119908119909
119898
isin 119876119898 ge 0and deg(119908) le V then 119897119888(119908) isin 119903119877 119897119888(119908)(119876 cap 119877) sube 119876 and V isa good number for all 119886 isin 119903119877
Proof Let V be minimal positive integer such that there exists1199081015840
= 1198940+1198941119909+sdot sdot sdot+119903119909
Visin 119877[119909 120590] and119898 ge 1with 119887minus119908
1015840
119909119898
isin 119876
and deg(1199081015840) = V It is clear that 119903 notin 119876 If 119888 = 120590(119903) minus 119903 notin 119876 put119892 = 120590(119908
1015840
) minus 1199081015840 and
119860 = sum
119894isinZ120590119894
(119888) 1198771
(18)
Thus 119860 is a right ideal of 119877 with 120590(119860) = 119860 and 119860 sube 119876 Byassumption 119860 = 119877 then 119903 = sum
119904
119894=0120590119902119894(119888)119897119894 where 119897
119894isin 1198771 and
119902119894isin Z Put
119905 = 1199081015840
minus
119904
sum
119894=0
120590119902119894(119892) 120590minusV
(119897119894) (19)
International Journal of Mathematics and Mathematical Sciences 3
Comparing the leading coefficients of 1199081015840 and
sum119904
119894=0120590119902119894(119892)120590minusV(119897119894) we have that
119887 minus 119905119909119898
isin 119876 deg (119905) le V minus 1 (20)
which contradicts theminimality of VTherefore 120590(119903)minus119903 isin 119876consequently 119903 isin
Suppose that 119903119902 notin Q for some 119902 isin 119877 cap 119876 Put 1198921015840
=
1199081015840
120590minusV(119902) isin 119876 using similar arguments as above we can have
a contradiction Hence 119903(119876 cap 119877) sube 119876If there exists 119908 isin 119903119877[119909 120590] with 119887 minus 119908119909
119895
isin 119876 119895 ge 0 anddeg(119908) le V then using similar arguments as above we canshow that lc(119908) isin 119903119877 and lc(119908)(119876 cap 119877) ⫅ 119876 Moreover if119886 isin 119903119877 put 119861 = 119886119877 + 119876 cap 119877 we have that 119861 is a right ideal of119877 with 120590(119861) = 119861 and 119861 sube 119876
By assumption 119861 = 119886119877 + 119876 cap 119877 = 119877 Thus 1199081015840
= 11988611990810158401015840
+
1199021015840 where 119908
10158401015840
isin 119877[119909 120590] deg(11990810158401015840) le deg(1199081015840) and 1199021015840
isin 119876Therefore 119887 minus 119886119908
10158401015840
119909119898
isin 119876 Consequently V is a good numberfor all 119886 isin 119903119877
Lemma 4 Let 119869 be a right ideal 119877 with 120590(119869) = 119869 119869 sube 119876 and119869 = 119877 such that for all sufficiently large 119899 there are 119891
119899isin 119869[119909 120590]
such that 119887 minus 119891119899119909119899
isin 119876 and deg(119891119899) le 119896 where 119876 is a right
ideal of 119877[119909 120590] and 119887 isin 119877[119909 120590] such that 119886 minus 119887119886 isin 119876 for every119886 isin 119877[119909 120590] Then there exists a positive integer V and 119903 isin
such that if 119908 isin 119903119877[119909 120590] with 119887 minus 119908119909119898
isin 119876 119898 ge 0 anddeg(119908) le V one has that 119897119888(119908) isin 119903119877 119897119888(119908)(119876 cap 119877) sube 119876 and Vis a good number for all 119886 isin 119903119877
Proof Let V be minimal positive integer such that for allsufficiently large119899 there are119891
119899isin 119869[119909 120590] such that 119887minus119891
119899119909119899
isin 119876
and deg(119891119899) le V Put
1199081015840
= 1198940+ 1198941119909 + sdot sdot sdot + 119894Vminus1119909
Vminus1+ 119903119909
Visin 119869 [119909 120590] (21)
with 119887 minus1199081015840
119909119898
isin 119876119898 ge 0 and deg(1199081015840) le V By Lemma 1 andminimality of V we have that 119903 notin 119876 Using the same ideas ofLemma 3 we have that 119903 isin and 119903(119876cap119877) sube 119876 Since 119903119877 sube 119869we have that the first part of lemma is satisfied
Let 119886 isin 119903119877 sube we denote by 119861 the right ideal of 119877
119861 = sum
119894isinZ
120590119894
(119886) 1198771
120590 (119861) = 119861 119861 sube 119869 119861 sube 119876 (22)
For sufficiently large 119899 there are 119892119899isin 119861[119909 120590] sube 119869[119909 120590] such
that 119887 minus 119892119899119909119899
isin 119876 and deg(119892119899) le V Put
119892119899= 1198881198990
+ 1198881198991
119909 + sdot sdot sdot + 119888119899V119909Visin 119861 [119909 120590] (23)
For every 0 le 119895 le V we have that 119888119899119895
= sum
119898119895
119894=0120590119902119899119894 (119886)119897119899119894
where119897119899119894
isin 1198771 and 119902
119899119894
isin Z Consequently
119888119899119895
=
119898119895
sum
119894=0
(120590119902119899119894 (119886) minus 119886) 119897
119899119894
+ 119886
119898119895
sum
119894=0
119897119899119894
(24)
Since 119886 isin 119903119877 we can write
119888119899119895
= 119904119899119895
+ 119903119899119895
119904119899119895
isin 119876 cap 119877 119903119899119895
isin 119877 (25)
Put ℎ119899= 1199031198990
+1199031198991
119909+ sdot sdot sdot + 119903119899V119909V thus 119887minus119886ℎ
119899119909119899
isin 119876 ThereforeV is a good number for all 119886 isin 119903119877
Lemma 5 Let 119876 be a right ideal of 119877[119909 120590] 119887 isin 119877[119909 120590] suchthat 119886minus 119887119886 isin 119876 for all 119886 isin 119877[119909 120590] and V is good number for all119886 isin 119903119877 where 119903 isin Assume that for every 119908 isin 119903119877[119909 120590] with119887 minus 119908119909
119898
isin 119876 119898 ge 0 and deg(119908) le V one has that 119897119888(119908) isin 119903119877
and 119897119888(119908)(119876 cap 119877) sube 119876 If there are 119901 and 1199011015840
isin 119903119877 with
(119901119877 + 119876 cap 119903119877) cap (1199011015840
119877 + 119876 cap 119903119877) sube 119876 (26)
then V minus 1 is a good number for 119903
Proof Since V is a good number for 119901 and 1199011015840 then for every
sufficiently large 119899 there are 119892119899isin 119901119877[119909 120590] and 119892
1015840
119899isin 1199011015840
119877[119909 120590]
such that
119887 minus 119892119899119909119899
isin 119876 (27)
119887 minus 1198921015840
119899119909119899
isin 119876 (28)
with deg(119892119899) deg(1198921015840
119899) le V Consider
119892119899= 1199011198990
+ 1199011198991
119909 + sdot sdot sdot + 119901119899V119909V 119901119899V
isin 119901119877
1198921015840
119899= 1199011015840
1198990
+ 1199011015840
1198991
119909 + sdot sdot sdot + 1199011015840
119899V119909V 1199011015840
119899Visin
1199011015840
119877
(29)
Since 119901119899V
minus 1199011015840
119899Visin 119876 then
119901119899V
isin (119901119877 + 119876 cap 119903119877) cap (1199011015840
119877 + 119876 cap 119903119877) sube 119876 (30)
a contradictionThus there exists sufficiently large 119894 isin N such that 119888 =
119901119894Vminus 1199011015840
119894Visin 119903119877 hence V is a good number for 119888 Then for all
sufficiently large 119899 there are ℎ119899isin 119877[119909 120590] such that 119887minus119888ℎ
119899119909119899
isin
119876 and deg(ℎ119899) le V We denote
ℎ119899= 1199031198990
+ 1199031198991
119909 + sdot sdot sdot + 119903119899V119909V (31)
Consider
119896119899= 119888ℎ119899+ (1198921015840
119894minus 119892119894) 120590minusV
(119903119899V) isin 119903119877 [119909 120590] (32)
Since 1198921015840
119894minus 119892119894isin 119876 then 119887 minus 119896
119899119909119899
isin 119876 Moreover
119896119899= 119888ℎ119899minus 119888119903119899V119909V+
Vminus1
sum
119895=0
(1199011015840
119894119895
minus 119901119894119895
) 119909119895
120590minusV
(119903119899V) (33)
Consequently V minus 1 is a good number for 119903
The following theorem extends ([1 Theorem 1])
Theorem 6 Let 119877 be a nil ring and let 119868 be a 120590-primitive idealin 119877[119909 120590] Then 119868 = 119868
1015840
[119909 120590] where 1198681015840 is an ideal 120590-invariant
of 119877
Proof Assume by contradiction that there are 1198860 1198861 119886
119896isin
119877 with
1198860+ 1198861119909 + sdot sdot sdot + 119886
119896119909119896
isin 119868 119886119896notin 119868 (34)
4 International Journal of Mathematics and Mathematical Sciences
Since 119868 is a 120590-primitive ideal in119877[119909 120590] there is a right ideal119876of119877[119909 120590]with 120590(119876) = 119876 and 119887 isin 119877[119909 120590] such that 119886minus119887119886 isin 119876
for all 119886 isin 119877[119909 120590] Moreover 119876 is a maximal in the set ofright ideals 120590-invariants and 119868 is the maximal ideal containedin 119876 We have that 119877[119909 120590]119909⫅119876 otherwise 119887 isin 119877 which isimpossible because119877 is a nil ring By definition of119876 it followsthat 119877[119909 120590]119909 + 119876 = 119877[119909 120590]
If 119887 minus ℎ119909119894
isin 119876 for some 119894 ge 0 with ℎ isin 119877[119909 120590] thendeg(ℎ) ge 1 In fact if ℎ isin 119877 let 119905 ge 1 be the minimal positiveinteger with respect to ℎ
119905
isin 119876 Thus (119887 minus ℎ119909119894
)120590minus119894
(ℎ119905minus1
) isin 119876Then 119887120590
minus119894
(ℎ119905minus1
) isin 119876 hence 120590minus119894
(ℎ119905minus1
) isin 119876 Consequentlyℎ119905minus1
isin 119876 a contradictionLet 119869 be a right ideal of 119877 with 120590(119869) = 119869 and 119869 sube 119876 We
have that 119869[119909 120590]119909+119876 = 119877[119909 120590]There exists119891 isin 119869[119909 120590] suchthat 119887 minus 119891119909 isin 119876 Consider
119880 = sum
119894isinZ
119869 [119909 120590] 120590119894
(119886119896) 1198771
[119909 120590] (35)
Since 119868 is an ideal 120590-prime and 119886119896
notin 119868 then 119880 sube 119868Consequently 119880 sube 119876 because 119868 is the maximal idealcontained in 119876 Then 119880
deg(119891)+ 119876 = 119877[119909 120590] There exists
1198921015840
isin 119880deg(119891) such that 119887 minus 119892
1015840
isin 119876 By Lemma 2 for every119894 ge deg(1198921015840) there are 119892
1015840
119894isin 119869[119909 120590] such that 119887 minus 119892
1015840
119894119909119894
isin 119876 anddeg(1198921015840
119894) lt 119896 Lemmas 3 and 4 imply that there are 119903
1015840
isin andV1015840 ge 1 such that if119908 isin 119903
1015840
119877[119909 120590]with 119887minus119908119909119898
isin 119876119898 ge 1 anddeg(119908) le V1015840 then lc(119908) isin
1199031015840
119877 and lc119908(119876cap119877) sube 119876 MoreoverV1015840 is a good number for all 119886 isin
1199031015840
119877 Let V beminimal such thatV is a good number for all 119886 isin
1199031015840
119877 We have that V le V1015840 Let119903 isin
1199031015840
119877 Since V is a good number for 119903 then for sufficientlylarge 119899 there are ℎ
119899isin 119877[119909 120590] such that
119887 minus 119903ℎ119899119909119899
isin 119876 deg (ℎ119899) le V (36)
Consider 119891119899= 119903ℎ119899 then 119887 minus 119891
119899119909119899
isin 119876 and deg(119891119899) le V For
some 119894 isin N there are 119891119894 119891119894+1
119891119894+119896
isin 119903119877[119909 120590] such that119887 minus 119891119895119909119895
isin 119876 deg(119891119895) le V and 119894 le 119895 le 119894 + 119896 Put
119891119895= 1199031198861198950
+ 1199031198861198951
119909 + sdot sdot sdot + 119903119886119895V119909V= 119892119895+ 119888119895119909V (37)
where 119892119895
= 1199031198861198950
+ 1199031198861198951
119909 + sdot sdot sdot + 119903119886119895Vminus1
119909Vminus1
isin 119903119877[119909 120590] and119888119895= 119903119886119895V Since deg(119891
119895) le V le V1015840 then 119888
119895notin 119876 Moreover
120590 (119888119895) minus 119888119895isin 119876 119888
119895(119876 cap 119877) sube 119876 (38)
Since 119877 is a nil ring consider 119890119895= 119888
119899119895
119895 where 119899
119895is a minimal
with respect to the condition 119888
119899119895
119895notin 119876 Thus 120590(119890
119895) minus 119890119895isin 119876 for
all 119894 ge 0 We have that
119891119895120590119895
(119890119895) = 119892119895120590119895
(119890119895) + 119888119895120590119895+V
(119890119895) 119909
V
= 119892119895120590119895
(119890119895) + 119888119895(120590119895+V
(119890119895) minus 119890119895) 119909
V
+ 119888119895119890119895119909V
(39)
Put 119905119895= 119892119895120590119895
(119890119895) isin 119903119877[119909 120590] Thus
119891119895120590119895
(119890119895) minus 119905119895isin 119876 deg (119905
119895) le V minus 1 (40)
for every 119894 le 119895 le 119894 + 119896 Since 119890119895isin 119903119877 sube
1199031015840
119877 if V minus 1 is not agood number for 119903 then Lemma 5 implies that
119894+119896
⋂
119895=1
(119890119895119877 + 119876 cap 119903119877) ⫅119876 (41)
In this case there exists 119904 isin ⋂119894+119896
119895=1(119890119895119877+119876cap119903119877) such that 119904 notin 119876
Consequently 119904minus 119890119895119889119895isin 119876cap119903119877 119889
119895isin 119877 and 119890
119895119889119895isin 119890119895119877 Then
119904 isin 119903119877 sube1199031015840
119877 Therefore V is a good number for 119904 Then forsufficiently large 119899 there are 119891
119899isin 119904119877[119909 120590] such that
119887 minus 119891119899119909119899
isin 119876 deg (119891119899) le V (42)
Let
119891119899=
V
sum
119895=0
119904119887119895119899
119909119895
(43)
Since 119887 minus 119891119895119909119895
isin 119876 119904 minus 119890119895119889119895
isin 119876 and 119890119895119889119895minus 119887119890119895119889119895
isin 119876then (119887 minus 119891
119895119909119895
)119890119895119889119895isin 119876 Thus 119887119890
119895119889119895minus 119891119895119909119895
119890119895119889119895isin 119876 hence
119904 minus 119891119895119909119895
119890119895119889119895isin 119876 for every 119894 le 119895 le 119894 + 119896
Let
119892119899=
V
sum
119895=0
119891119894+Vminus119895120590
119894+Vminus119895(119890119894+Vminus119895119889119894+Vminus119895119887119895
119899
) 119909119894+V
isin 119903119877 [119909 120590] (44)
We have that 119891119899minus119892119899isin 119876 Thus 119887minus119892
119899119909119899
= (119887minus119891119899119909119899
) + (119891119899minus
119892119899)119909119899
isin 119876 Put
ℎ119899=
V
sum
119895=0
119905119894+Vminus119895120590
119894+Vminus119895(119889119894+Vminus119895119887119895
119899
) isin 119903119877 [119909 120590] (45)
We can write 119887 minus ℎ119899119909119894+V+119899 as
119887 minus
V
sum
119895=0
(119905119894+Vminus119895 minus 119891
119894+Vminus119895120590119894+Vminus119895
(119890119894+Vminus119895)) 120590
119894+Vminus119895(119889119894+Vminus119895119887119895
119899
)
sdot 119909119894+V+119899
minus 119892119899119909119899
(46)
Thus for all sufficient large 119899
119887 minus ℎ119899119909119894+V+119899
isin 119876 deg (ℎ119899) le V minus 1 (47)
Then V minus 1 is a good number for all 119903 isin1199031015840
119877 This contradictsthe minimality of V
Recall that the 120590-Jacobson radical 119869120590(119877) of a ring 119877 is
defined as the intersection of all 120590-primitive ideals of 119877 Aring 119877 is a 120590-Jacobson radical if 119869
120590(119877) = 119877
Theorem 7 Let 119877 be a nil ring and let 119868 be an ideal of 119877[119909 120590]Consider 119868 the ideal of119877 generated by coefficients of polynomialfrom 119868 Then 119877[119909 120590]119868[119909 120590] is 120590-Jacobson radical if and onlyif 119877[119909 120590]119868 is 120590-Jacobson radical
International Journal of Mathematics and Mathematical Sciences 5
Proof Assume by contradiction that 119877[119909 120590]119868 is not 120590-Jacobson radical Then there is a 120590-primitive ideal 119875 of119877[119909 120590]119868 such that 119875 = 119877[119909 120590]119868 We have that there is anideal 119870 of 119877[119909 120590] such that 119875 = 119870119868 Therefore 119870 is a 120590-primitive ideal of 119877[119909 120590] By Theorem 6 there is an ideal 119875of 119877 such that119870 = 119875[119909 120590] It is clear that 119868 sube 119875 Since
(119877 [119909 120590] 119868 [119909 120590])
(119875 [119909 120590] 119868 [119909 120590])
≃
119877 [119909 120590]
119870
(48)
then 119875[119909 120590]119868[119909 120590] is a 120590-primitive ideal a contradictionUsing the fact that 119868 sube 119868[119909 120590] the converse follows
Corollary 8 If 119877 is a nil ring then the polynomial ring of typeautomorphism 119877[119909 120590] can not be homomorphically mappedonto a 120590-simple 120590-primitive ring
Competing Interests
The author declares that they have no competing interests
References
[1] A Smoktunowicz ldquoOn primitive ideals in polynomial ringsover nil ringsrdquo Algebras and Representation Theory vol 8 no1 pp 69ndash73 2005
[2] E Cisneros M Ferrero and M I Conzles ldquoPrime ideals ofskew polynomial rings and skew laurent polynomial ringsrdquoMathematical Journal of Okayama University vol 32 pp 61ndash721990
[3] NDivinskyRings and Radicals Allen andUnwin London UK1965
[4] T Y Lam A First Course in Noncommutative Rings GraduateTexts in Mathematics Springer New York NY USA 1991
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International Journal of Mathematics and Mathematical Sciences 3
Comparing the leading coefficients of 1199081015840 and
sum119904
119894=0120590119902119894(119892)120590minusV(119897119894) we have that
119887 minus 119905119909119898
isin 119876 deg (119905) le V minus 1 (20)
which contradicts theminimality of VTherefore 120590(119903)minus119903 isin 119876consequently 119903 isin
Suppose that 119903119902 notin Q for some 119902 isin 119877 cap 119876 Put 1198921015840
=
1199081015840
120590minusV(119902) isin 119876 using similar arguments as above we can have
a contradiction Hence 119903(119876 cap 119877) sube 119876If there exists 119908 isin 119903119877[119909 120590] with 119887 minus 119908119909
119895
isin 119876 119895 ge 0 anddeg(119908) le V then using similar arguments as above we canshow that lc(119908) isin 119903119877 and lc(119908)(119876 cap 119877) ⫅ 119876 Moreover if119886 isin 119903119877 put 119861 = 119886119877 + 119876 cap 119877 we have that 119861 is a right ideal of119877 with 120590(119861) = 119861 and 119861 sube 119876
By assumption 119861 = 119886119877 + 119876 cap 119877 = 119877 Thus 1199081015840
= 11988611990810158401015840
+
1199021015840 where 119908
10158401015840
isin 119877[119909 120590] deg(11990810158401015840) le deg(1199081015840) and 1199021015840
isin 119876Therefore 119887 minus 119886119908
10158401015840
119909119898
isin 119876 Consequently V is a good numberfor all 119886 isin 119903119877
Lemma 4 Let 119869 be a right ideal 119877 with 120590(119869) = 119869 119869 sube 119876 and119869 = 119877 such that for all sufficiently large 119899 there are 119891
119899isin 119869[119909 120590]
such that 119887 minus 119891119899119909119899
isin 119876 and deg(119891119899) le 119896 where 119876 is a right
ideal of 119877[119909 120590] and 119887 isin 119877[119909 120590] such that 119886 minus 119887119886 isin 119876 for every119886 isin 119877[119909 120590] Then there exists a positive integer V and 119903 isin
such that if 119908 isin 119903119877[119909 120590] with 119887 minus 119908119909119898
isin 119876 119898 ge 0 anddeg(119908) le V one has that 119897119888(119908) isin 119903119877 119897119888(119908)(119876 cap 119877) sube 119876 and Vis a good number for all 119886 isin 119903119877
Proof Let V be minimal positive integer such that for allsufficiently large119899 there are119891
119899isin 119869[119909 120590] such that 119887minus119891
119899119909119899
isin 119876
and deg(119891119899) le V Put
1199081015840
= 1198940+ 1198941119909 + sdot sdot sdot + 119894Vminus1119909
Vminus1+ 119903119909
Visin 119869 [119909 120590] (21)
with 119887 minus1199081015840
119909119898
isin 119876119898 ge 0 and deg(1199081015840) le V By Lemma 1 andminimality of V we have that 119903 notin 119876 Using the same ideas ofLemma 3 we have that 119903 isin and 119903(119876cap119877) sube 119876 Since 119903119877 sube 119869we have that the first part of lemma is satisfied
Let 119886 isin 119903119877 sube we denote by 119861 the right ideal of 119877
119861 = sum
119894isinZ
120590119894
(119886) 1198771
120590 (119861) = 119861 119861 sube 119869 119861 sube 119876 (22)
For sufficiently large 119899 there are 119892119899isin 119861[119909 120590] sube 119869[119909 120590] such
that 119887 minus 119892119899119909119899
isin 119876 and deg(119892119899) le V Put
119892119899= 1198881198990
+ 1198881198991
119909 + sdot sdot sdot + 119888119899V119909Visin 119861 [119909 120590] (23)
For every 0 le 119895 le V we have that 119888119899119895
= sum
119898119895
119894=0120590119902119899119894 (119886)119897119899119894
where119897119899119894
isin 1198771 and 119902
119899119894
isin Z Consequently
119888119899119895
=
119898119895
sum
119894=0
(120590119902119899119894 (119886) minus 119886) 119897
119899119894
+ 119886
119898119895
sum
119894=0
119897119899119894
(24)
Since 119886 isin 119903119877 we can write
119888119899119895
= 119904119899119895
+ 119903119899119895
119904119899119895
isin 119876 cap 119877 119903119899119895
isin 119877 (25)
Put ℎ119899= 1199031198990
+1199031198991
119909+ sdot sdot sdot + 119903119899V119909V thus 119887minus119886ℎ
119899119909119899
isin 119876 ThereforeV is a good number for all 119886 isin 119903119877
Lemma 5 Let 119876 be a right ideal of 119877[119909 120590] 119887 isin 119877[119909 120590] suchthat 119886minus 119887119886 isin 119876 for all 119886 isin 119877[119909 120590] and V is good number for all119886 isin 119903119877 where 119903 isin Assume that for every 119908 isin 119903119877[119909 120590] with119887 minus 119908119909
119898
isin 119876 119898 ge 0 and deg(119908) le V one has that 119897119888(119908) isin 119903119877
and 119897119888(119908)(119876 cap 119877) sube 119876 If there are 119901 and 1199011015840
isin 119903119877 with
(119901119877 + 119876 cap 119903119877) cap (1199011015840
119877 + 119876 cap 119903119877) sube 119876 (26)
then V minus 1 is a good number for 119903
Proof Since V is a good number for 119901 and 1199011015840 then for every
sufficiently large 119899 there are 119892119899isin 119901119877[119909 120590] and 119892
1015840
119899isin 1199011015840
119877[119909 120590]
such that
119887 minus 119892119899119909119899
isin 119876 (27)
119887 minus 1198921015840
119899119909119899
isin 119876 (28)
with deg(119892119899) deg(1198921015840
119899) le V Consider
119892119899= 1199011198990
+ 1199011198991
119909 + sdot sdot sdot + 119901119899V119909V 119901119899V
isin 119901119877
1198921015840
119899= 1199011015840
1198990
+ 1199011015840
1198991
119909 + sdot sdot sdot + 1199011015840
119899V119909V 1199011015840
119899Visin
1199011015840
119877
(29)
Since 119901119899V
minus 1199011015840
119899Visin 119876 then
119901119899V
isin (119901119877 + 119876 cap 119903119877) cap (1199011015840
119877 + 119876 cap 119903119877) sube 119876 (30)
a contradictionThus there exists sufficiently large 119894 isin N such that 119888 =
119901119894Vminus 1199011015840
119894Visin 119903119877 hence V is a good number for 119888 Then for all
sufficiently large 119899 there are ℎ119899isin 119877[119909 120590] such that 119887minus119888ℎ
119899119909119899
isin
119876 and deg(ℎ119899) le V We denote
ℎ119899= 1199031198990
+ 1199031198991
119909 + sdot sdot sdot + 119903119899V119909V (31)
Consider
119896119899= 119888ℎ119899+ (1198921015840
119894minus 119892119894) 120590minusV
(119903119899V) isin 119903119877 [119909 120590] (32)
Since 1198921015840
119894minus 119892119894isin 119876 then 119887 minus 119896
119899119909119899
isin 119876 Moreover
119896119899= 119888ℎ119899minus 119888119903119899V119909V+
Vminus1
sum
119895=0
(1199011015840
119894119895
minus 119901119894119895
) 119909119895
120590minusV
(119903119899V) (33)
Consequently V minus 1 is a good number for 119903
The following theorem extends ([1 Theorem 1])
Theorem 6 Let 119877 be a nil ring and let 119868 be a 120590-primitive idealin 119877[119909 120590] Then 119868 = 119868
1015840
[119909 120590] where 1198681015840 is an ideal 120590-invariant
of 119877
Proof Assume by contradiction that there are 1198860 1198861 119886
119896isin
119877 with
1198860+ 1198861119909 + sdot sdot sdot + 119886
119896119909119896
isin 119868 119886119896notin 119868 (34)
4 International Journal of Mathematics and Mathematical Sciences
Since 119868 is a 120590-primitive ideal in119877[119909 120590] there is a right ideal119876of119877[119909 120590]with 120590(119876) = 119876 and 119887 isin 119877[119909 120590] such that 119886minus119887119886 isin 119876
for all 119886 isin 119877[119909 120590] Moreover 119876 is a maximal in the set ofright ideals 120590-invariants and 119868 is the maximal ideal containedin 119876 We have that 119877[119909 120590]119909⫅119876 otherwise 119887 isin 119877 which isimpossible because119877 is a nil ring By definition of119876 it followsthat 119877[119909 120590]119909 + 119876 = 119877[119909 120590]
If 119887 minus ℎ119909119894
isin 119876 for some 119894 ge 0 with ℎ isin 119877[119909 120590] thendeg(ℎ) ge 1 In fact if ℎ isin 119877 let 119905 ge 1 be the minimal positiveinteger with respect to ℎ
119905
isin 119876 Thus (119887 minus ℎ119909119894
)120590minus119894
(ℎ119905minus1
) isin 119876Then 119887120590
minus119894
(ℎ119905minus1
) isin 119876 hence 120590minus119894
(ℎ119905minus1
) isin 119876 Consequentlyℎ119905minus1
isin 119876 a contradictionLet 119869 be a right ideal of 119877 with 120590(119869) = 119869 and 119869 sube 119876 We
have that 119869[119909 120590]119909+119876 = 119877[119909 120590]There exists119891 isin 119869[119909 120590] suchthat 119887 minus 119891119909 isin 119876 Consider
119880 = sum
119894isinZ
119869 [119909 120590] 120590119894
(119886119896) 1198771
[119909 120590] (35)
Since 119868 is an ideal 120590-prime and 119886119896
notin 119868 then 119880 sube 119868Consequently 119880 sube 119876 because 119868 is the maximal idealcontained in 119876 Then 119880
deg(119891)+ 119876 = 119877[119909 120590] There exists
1198921015840
isin 119880deg(119891) such that 119887 minus 119892
1015840
isin 119876 By Lemma 2 for every119894 ge deg(1198921015840) there are 119892
1015840
119894isin 119869[119909 120590] such that 119887 minus 119892
1015840
119894119909119894
isin 119876 anddeg(1198921015840
119894) lt 119896 Lemmas 3 and 4 imply that there are 119903
1015840
isin andV1015840 ge 1 such that if119908 isin 119903
1015840
119877[119909 120590]with 119887minus119908119909119898
isin 119876119898 ge 1 anddeg(119908) le V1015840 then lc(119908) isin
1199031015840
119877 and lc119908(119876cap119877) sube 119876 MoreoverV1015840 is a good number for all 119886 isin
1199031015840
119877 Let V beminimal such thatV is a good number for all 119886 isin
1199031015840
119877 We have that V le V1015840 Let119903 isin
1199031015840
119877 Since V is a good number for 119903 then for sufficientlylarge 119899 there are ℎ
119899isin 119877[119909 120590] such that
119887 minus 119903ℎ119899119909119899
isin 119876 deg (ℎ119899) le V (36)
Consider 119891119899= 119903ℎ119899 then 119887 minus 119891
119899119909119899
isin 119876 and deg(119891119899) le V For
some 119894 isin N there are 119891119894 119891119894+1
119891119894+119896
isin 119903119877[119909 120590] such that119887 minus 119891119895119909119895
isin 119876 deg(119891119895) le V and 119894 le 119895 le 119894 + 119896 Put
119891119895= 1199031198861198950
+ 1199031198861198951
119909 + sdot sdot sdot + 119903119886119895V119909V= 119892119895+ 119888119895119909V (37)
where 119892119895
= 1199031198861198950
+ 1199031198861198951
119909 + sdot sdot sdot + 119903119886119895Vminus1
119909Vminus1
isin 119903119877[119909 120590] and119888119895= 119903119886119895V Since deg(119891
119895) le V le V1015840 then 119888
119895notin 119876 Moreover
120590 (119888119895) minus 119888119895isin 119876 119888
119895(119876 cap 119877) sube 119876 (38)
Since 119877 is a nil ring consider 119890119895= 119888
119899119895
119895 where 119899
119895is a minimal
with respect to the condition 119888
119899119895
119895notin 119876 Thus 120590(119890
119895) minus 119890119895isin 119876 for
all 119894 ge 0 We have that
119891119895120590119895
(119890119895) = 119892119895120590119895
(119890119895) + 119888119895120590119895+V
(119890119895) 119909
V
= 119892119895120590119895
(119890119895) + 119888119895(120590119895+V
(119890119895) minus 119890119895) 119909
V
+ 119888119895119890119895119909V
(39)
Put 119905119895= 119892119895120590119895
(119890119895) isin 119903119877[119909 120590] Thus
119891119895120590119895
(119890119895) minus 119905119895isin 119876 deg (119905
119895) le V minus 1 (40)
for every 119894 le 119895 le 119894 + 119896 Since 119890119895isin 119903119877 sube
1199031015840
119877 if V minus 1 is not agood number for 119903 then Lemma 5 implies that
119894+119896
⋂
119895=1
(119890119895119877 + 119876 cap 119903119877) ⫅119876 (41)
In this case there exists 119904 isin ⋂119894+119896
119895=1(119890119895119877+119876cap119903119877) such that 119904 notin 119876
Consequently 119904minus 119890119895119889119895isin 119876cap119903119877 119889
119895isin 119877 and 119890
119895119889119895isin 119890119895119877 Then
119904 isin 119903119877 sube1199031015840
119877 Therefore V is a good number for 119904 Then forsufficiently large 119899 there are 119891
119899isin 119904119877[119909 120590] such that
119887 minus 119891119899119909119899
isin 119876 deg (119891119899) le V (42)
Let
119891119899=
V
sum
119895=0
119904119887119895119899
119909119895
(43)
Since 119887 minus 119891119895119909119895
isin 119876 119904 minus 119890119895119889119895
isin 119876 and 119890119895119889119895minus 119887119890119895119889119895
isin 119876then (119887 minus 119891
119895119909119895
)119890119895119889119895isin 119876 Thus 119887119890
119895119889119895minus 119891119895119909119895
119890119895119889119895isin 119876 hence
119904 minus 119891119895119909119895
119890119895119889119895isin 119876 for every 119894 le 119895 le 119894 + 119896
Let
119892119899=
V
sum
119895=0
119891119894+Vminus119895120590
119894+Vminus119895(119890119894+Vminus119895119889119894+Vminus119895119887119895
119899
) 119909119894+V
isin 119903119877 [119909 120590] (44)
We have that 119891119899minus119892119899isin 119876 Thus 119887minus119892
119899119909119899
= (119887minus119891119899119909119899
) + (119891119899minus
119892119899)119909119899
isin 119876 Put
ℎ119899=
V
sum
119895=0
119905119894+Vminus119895120590
119894+Vminus119895(119889119894+Vminus119895119887119895
119899
) isin 119903119877 [119909 120590] (45)
We can write 119887 minus ℎ119899119909119894+V+119899 as
119887 minus
V
sum
119895=0
(119905119894+Vminus119895 minus 119891
119894+Vminus119895120590119894+Vminus119895
(119890119894+Vminus119895)) 120590
119894+Vminus119895(119889119894+Vminus119895119887119895
119899
)
sdot 119909119894+V+119899
minus 119892119899119909119899
(46)
Thus for all sufficient large 119899
119887 minus ℎ119899119909119894+V+119899
isin 119876 deg (ℎ119899) le V minus 1 (47)
Then V minus 1 is a good number for all 119903 isin1199031015840
119877 This contradictsthe minimality of V
Recall that the 120590-Jacobson radical 119869120590(119877) of a ring 119877 is
defined as the intersection of all 120590-primitive ideals of 119877 Aring 119877 is a 120590-Jacobson radical if 119869
120590(119877) = 119877
Theorem 7 Let 119877 be a nil ring and let 119868 be an ideal of 119877[119909 120590]Consider 119868 the ideal of119877 generated by coefficients of polynomialfrom 119868 Then 119877[119909 120590]119868[119909 120590] is 120590-Jacobson radical if and onlyif 119877[119909 120590]119868 is 120590-Jacobson radical
International Journal of Mathematics and Mathematical Sciences 5
Proof Assume by contradiction that 119877[119909 120590]119868 is not 120590-Jacobson radical Then there is a 120590-primitive ideal 119875 of119877[119909 120590]119868 such that 119875 = 119877[119909 120590]119868 We have that there is anideal 119870 of 119877[119909 120590] such that 119875 = 119870119868 Therefore 119870 is a 120590-primitive ideal of 119877[119909 120590] By Theorem 6 there is an ideal 119875of 119877 such that119870 = 119875[119909 120590] It is clear that 119868 sube 119875 Since
(119877 [119909 120590] 119868 [119909 120590])
(119875 [119909 120590] 119868 [119909 120590])
≃
119877 [119909 120590]
119870
(48)
then 119875[119909 120590]119868[119909 120590] is a 120590-primitive ideal a contradictionUsing the fact that 119868 sube 119868[119909 120590] the converse follows
Corollary 8 If 119877 is a nil ring then the polynomial ring of typeautomorphism 119877[119909 120590] can not be homomorphically mappedonto a 120590-simple 120590-primitive ring
Competing Interests
The author declares that they have no competing interests
References
[1] A Smoktunowicz ldquoOn primitive ideals in polynomial ringsover nil ringsrdquo Algebras and Representation Theory vol 8 no1 pp 69ndash73 2005
[2] E Cisneros M Ferrero and M I Conzles ldquoPrime ideals ofskew polynomial rings and skew laurent polynomial ringsrdquoMathematical Journal of Okayama University vol 32 pp 61ndash721990
[3] NDivinskyRings and Radicals Allen andUnwin London UK1965
[4] T Y Lam A First Course in Noncommutative Rings GraduateTexts in Mathematics Springer New York NY USA 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Mathematics and Mathematical Sciences
Since 119868 is a 120590-primitive ideal in119877[119909 120590] there is a right ideal119876of119877[119909 120590]with 120590(119876) = 119876 and 119887 isin 119877[119909 120590] such that 119886minus119887119886 isin 119876
for all 119886 isin 119877[119909 120590] Moreover 119876 is a maximal in the set ofright ideals 120590-invariants and 119868 is the maximal ideal containedin 119876 We have that 119877[119909 120590]119909⫅119876 otherwise 119887 isin 119877 which isimpossible because119877 is a nil ring By definition of119876 it followsthat 119877[119909 120590]119909 + 119876 = 119877[119909 120590]
If 119887 minus ℎ119909119894
isin 119876 for some 119894 ge 0 with ℎ isin 119877[119909 120590] thendeg(ℎ) ge 1 In fact if ℎ isin 119877 let 119905 ge 1 be the minimal positiveinteger with respect to ℎ
119905
isin 119876 Thus (119887 minus ℎ119909119894
)120590minus119894
(ℎ119905minus1
) isin 119876Then 119887120590
minus119894
(ℎ119905minus1
) isin 119876 hence 120590minus119894
(ℎ119905minus1
) isin 119876 Consequentlyℎ119905minus1
isin 119876 a contradictionLet 119869 be a right ideal of 119877 with 120590(119869) = 119869 and 119869 sube 119876 We
have that 119869[119909 120590]119909+119876 = 119877[119909 120590]There exists119891 isin 119869[119909 120590] suchthat 119887 minus 119891119909 isin 119876 Consider
119880 = sum
119894isinZ
119869 [119909 120590] 120590119894
(119886119896) 1198771
[119909 120590] (35)
Since 119868 is an ideal 120590-prime and 119886119896
notin 119868 then 119880 sube 119868Consequently 119880 sube 119876 because 119868 is the maximal idealcontained in 119876 Then 119880
deg(119891)+ 119876 = 119877[119909 120590] There exists
1198921015840
isin 119880deg(119891) such that 119887 minus 119892
1015840
isin 119876 By Lemma 2 for every119894 ge deg(1198921015840) there are 119892
1015840
119894isin 119869[119909 120590] such that 119887 minus 119892
1015840
119894119909119894
isin 119876 anddeg(1198921015840
119894) lt 119896 Lemmas 3 and 4 imply that there are 119903
1015840
isin andV1015840 ge 1 such that if119908 isin 119903
1015840
119877[119909 120590]with 119887minus119908119909119898
isin 119876119898 ge 1 anddeg(119908) le V1015840 then lc(119908) isin
1199031015840
119877 and lc119908(119876cap119877) sube 119876 MoreoverV1015840 is a good number for all 119886 isin
1199031015840
119877 Let V beminimal such thatV is a good number for all 119886 isin
1199031015840
119877 We have that V le V1015840 Let119903 isin
1199031015840
119877 Since V is a good number for 119903 then for sufficientlylarge 119899 there are ℎ
119899isin 119877[119909 120590] such that
119887 minus 119903ℎ119899119909119899
isin 119876 deg (ℎ119899) le V (36)
Consider 119891119899= 119903ℎ119899 then 119887 minus 119891
119899119909119899
isin 119876 and deg(119891119899) le V For
some 119894 isin N there are 119891119894 119891119894+1
119891119894+119896
isin 119903119877[119909 120590] such that119887 minus 119891119895119909119895
isin 119876 deg(119891119895) le V and 119894 le 119895 le 119894 + 119896 Put
119891119895= 1199031198861198950
+ 1199031198861198951
119909 + sdot sdot sdot + 119903119886119895V119909V= 119892119895+ 119888119895119909V (37)
where 119892119895
= 1199031198861198950
+ 1199031198861198951
119909 + sdot sdot sdot + 119903119886119895Vminus1
119909Vminus1
isin 119903119877[119909 120590] and119888119895= 119903119886119895V Since deg(119891
119895) le V le V1015840 then 119888
119895notin 119876 Moreover
120590 (119888119895) minus 119888119895isin 119876 119888
119895(119876 cap 119877) sube 119876 (38)
Since 119877 is a nil ring consider 119890119895= 119888
119899119895
119895 where 119899
119895is a minimal
with respect to the condition 119888
119899119895
119895notin 119876 Thus 120590(119890
119895) minus 119890119895isin 119876 for
all 119894 ge 0 We have that
119891119895120590119895
(119890119895) = 119892119895120590119895
(119890119895) + 119888119895120590119895+V
(119890119895) 119909
V
= 119892119895120590119895
(119890119895) + 119888119895(120590119895+V
(119890119895) minus 119890119895) 119909
V
+ 119888119895119890119895119909V
(39)
Put 119905119895= 119892119895120590119895
(119890119895) isin 119903119877[119909 120590] Thus
119891119895120590119895
(119890119895) minus 119905119895isin 119876 deg (119905
119895) le V minus 1 (40)
for every 119894 le 119895 le 119894 + 119896 Since 119890119895isin 119903119877 sube
1199031015840
119877 if V minus 1 is not agood number for 119903 then Lemma 5 implies that
119894+119896
⋂
119895=1
(119890119895119877 + 119876 cap 119903119877) ⫅119876 (41)
In this case there exists 119904 isin ⋂119894+119896
119895=1(119890119895119877+119876cap119903119877) such that 119904 notin 119876
Consequently 119904minus 119890119895119889119895isin 119876cap119903119877 119889
119895isin 119877 and 119890
119895119889119895isin 119890119895119877 Then
119904 isin 119903119877 sube1199031015840
119877 Therefore V is a good number for 119904 Then forsufficiently large 119899 there are 119891
119899isin 119904119877[119909 120590] such that
119887 minus 119891119899119909119899
isin 119876 deg (119891119899) le V (42)
Let
119891119899=
V
sum
119895=0
119904119887119895119899
119909119895
(43)
Since 119887 minus 119891119895119909119895
isin 119876 119904 minus 119890119895119889119895
isin 119876 and 119890119895119889119895minus 119887119890119895119889119895
isin 119876then (119887 minus 119891
119895119909119895
)119890119895119889119895isin 119876 Thus 119887119890
119895119889119895minus 119891119895119909119895
119890119895119889119895isin 119876 hence
119904 minus 119891119895119909119895
119890119895119889119895isin 119876 for every 119894 le 119895 le 119894 + 119896
Let
119892119899=
V
sum
119895=0
119891119894+Vminus119895120590
119894+Vminus119895(119890119894+Vminus119895119889119894+Vminus119895119887119895
119899
) 119909119894+V
isin 119903119877 [119909 120590] (44)
We have that 119891119899minus119892119899isin 119876 Thus 119887minus119892
119899119909119899
= (119887minus119891119899119909119899
) + (119891119899minus
119892119899)119909119899
isin 119876 Put
ℎ119899=
V
sum
119895=0
119905119894+Vminus119895120590
119894+Vminus119895(119889119894+Vminus119895119887119895
119899
) isin 119903119877 [119909 120590] (45)
We can write 119887 minus ℎ119899119909119894+V+119899 as
119887 minus
V
sum
119895=0
(119905119894+Vminus119895 minus 119891
119894+Vminus119895120590119894+Vminus119895
(119890119894+Vminus119895)) 120590
119894+Vminus119895(119889119894+Vminus119895119887119895
119899
)
sdot 119909119894+V+119899
minus 119892119899119909119899
(46)
Thus for all sufficient large 119899
119887 minus ℎ119899119909119894+V+119899
isin 119876 deg (ℎ119899) le V minus 1 (47)
Then V minus 1 is a good number for all 119903 isin1199031015840
119877 This contradictsthe minimality of V
Recall that the 120590-Jacobson radical 119869120590(119877) of a ring 119877 is
defined as the intersection of all 120590-primitive ideals of 119877 Aring 119877 is a 120590-Jacobson radical if 119869
120590(119877) = 119877
Theorem 7 Let 119877 be a nil ring and let 119868 be an ideal of 119877[119909 120590]Consider 119868 the ideal of119877 generated by coefficients of polynomialfrom 119868 Then 119877[119909 120590]119868[119909 120590] is 120590-Jacobson radical if and onlyif 119877[119909 120590]119868 is 120590-Jacobson radical
International Journal of Mathematics and Mathematical Sciences 5
Proof Assume by contradiction that 119877[119909 120590]119868 is not 120590-Jacobson radical Then there is a 120590-primitive ideal 119875 of119877[119909 120590]119868 such that 119875 = 119877[119909 120590]119868 We have that there is anideal 119870 of 119877[119909 120590] such that 119875 = 119870119868 Therefore 119870 is a 120590-primitive ideal of 119877[119909 120590] By Theorem 6 there is an ideal 119875of 119877 such that119870 = 119875[119909 120590] It is clear that 119868 sube 119875 Since
(119877 [119909 120590] 119868 [119909 120590])
(119875 [119909 120590] 119868 [119909 120590])
≃
119877 [119909 120590]
119870
(48)
then 119875[119909 120590]119868[119909 120590] is a 120590-primitive ideal a contradictionUsing the fact that 119868 sube 119868[119909 120590] the converse follows
Corollary 8 If 119877 is a nil ring then the polynomial ring of typeautomorphism 119877[119909 120590] can not be homomorphically mappedonto a 120590-simple 120590-primitive ring
Competing Interests
The author declares that they have no competing interests
References
[1] A Smoktunowicz ldquoOn primitive ideals in polynomial ringsover nil ringsrdquo Algebras and Representation Theory vol 8 no1 pp 69ndash73 2005
[2] E Cisneros M Ferrero and M I Conzles ldquoPrime ideals ofskew polynomial rings and skew laurent polynomial ringsrdquoMathematical Journal of Okayama University vol 32 pp 61ndash721990
[3] NDivinskyRings and Radicals Allen andUnwin London UK1965
[4] T Y Lam A First Course in Noncommutative Rings GraduateTexts in Mathematics Springer New York NY USA 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Mathematics and Mathematical Sciences 5
Proof Assume by contradiction that 119877[119909 120590]119868 is not 120590-Jacobson radical Then there is a 120590-primitive ideal 119875 of119877[119909 120590]119868 such that 119875 = 119877[119909 120590]119868 We have that there is anideal 119870 of 119877[119909 120590] such that 119875 = 119870119868 Therefore 119870 is a 120590-primitive ideal of 119877[119909 120590] By Theorem 6 there is an ideal 119875of 119877 such that119870 = 119875[119909 120590] It is clear that 119868 sube 119875 Since
(119877 [119909 120590] 119868 [119909 120590])
(119875 [119909 120590] 119868 [119909 120590])
≃
119877 [119909 120590]
119870
(48)
then 119875[119909 120590]119868[119909 120590] is a 120590-primitive ideal a contradictionUsing the fact that 119868 sube 119868[119909 120590] the converse follows
Corollary 8 If 119877 is a nil ring then the polynomial ring of typeautomorphism 119877[119909 120590] can not be homomorphically mappedonto a 120590-simple 120590-primitive ring
Competing Interests
The author declares that they have no competing interests
References
[1] A Smoktunowicz ldquoOn primitive ideals in polynomial ringsover nil ringsrdquo Algebras and Representation Theory vol 8 no1 pp 69ndash73 2005
[2] E Cisneros M Ferrero and M I Conzles ldquoPrime ideals ofskew polynomial rings and skew laurent polynomial ringsrdquoMathematical Journal of Okayama University vol 32 pp 61ndash721990
[3] NDivinskyRings and Radicals Allen andUnwin London UK1965
[4] T Y Lam A First Course in Noncommutative Rings GraduateTexts in Mathematics Springer New York NY USA 1991
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of