Results in Mathematics Vol. 29 (1996)

0378-6218/96/020125-12$1.50+0.20/0 (c) Birkhiiuser Verlag, Basel

Boolean Orthogonalities For Near-rings

Gordoll MasoI!

Abstract

Cornish has dev,'loped a tllPory of Boolean orthogonalitif's for sP!,s with an associated alg<'­braic closure system of "ideals", and applied it to redncf'd rings and s(,ll1iprilllP rings. In this paper we apply t1w theory to Iwar-rings and in particular to :1-sf'lllipriIllP Iwar-rings. As one (,01l8('qll('nc(', we ident.ify sorn" Iwar-rings whos{' :l-sellliprime i,kals aI'(' int,'rsections of 3-prim" ideals. In the final sect,ion, we discuss lo('al ideals and normality conJitions for lIear-rings wit.h a Boolean orthogonalit.y.

I!)!ll AMS Classification: HjY;lO

l\Py wonk Near-rin!!;, Boolpan orthogonalit.y, a-primp idpal.

INTRODUCTION. Tltroll!!;ltollt. this papPf, R will denotE' a zpro-synllIIPt.ric right. IIpar-ring. An

idpalf of R is callf'd primp if WllPllPVN II and B are ideals slIch that AB <;; I tlH'n A <;; / or B <;; I.

(HpfP as IIs11al AB = {ailia E It II E IJ}.) All iriE'all' with thf' propprty aRb <;; P => II. E P or I) E P

has bePll callpd type one primp [IJ or :1-prilllP [5]. WI' shall liSP tllf' lat.tpr tprlllinology. Clearly a

:l-primp idpa.l is prilllP hilt IIntil qllit.!' rpcPlitly lIot much was known about thp rpvprsp implication.

HoweVpf this was rectified in [1]. An idpal I is sPlltiprilllP (:l-spmiprinw) if wheupver .12 <;; I for

SOlTle ideal .I, tlwn .J <;; I (rpsppctiveiy II.IlI1. <;; J => fl E f). It has bpPll known for some timp (8pe

e.g. [10, p. fiXJ) that f is sE'liliprilllP ifF it is an int.Prsprtioll of primp idpals. Thp COf]'PSI)(JlHlin!!; res lIlt

for ;1-sellliprirnp idpills rPllIains pillsivp. Onp COIISP(I'H'ncp of thp thpory dpv('loppd ill this pappr is

to provide a class of ('xalllpiPs ill which :1-sPlllipfil!lP idpa.ls af(' illt.Pfspct.ions of :1-prilI1p idpals. A

fnrtlwr class of ('X a III pIps, ohtainpd ind!'pl'll<if'nt.iy in [2J, is also IlH'ntionp(1.

Standard Ilotations and H'snit.s 011 llear-rings call 1)(' fOllnd in [IOJ. WI' sltallns!' [:r] to dpllotp

t.hp idpa.] !!;(,IlPrat('d by an ('II'IIIPI1t. :/: E fl.

j. BOOLEAN ORTHOGONALITIES.

III [:l] (;orllish dpvploppd a. t.iH'ory of Boolean orthogonalities for a 8('1. S having a di8tillguishpd

('lplllPnt 0 and all algphraic clOSllf(, systPlIl J of "idpals" sllch t.hat n .I = {O}. In tbis s('dion WP .iE,7

apply Corllish's rpsllits t.o TH'ar-rill!!;s. W(' first. record SOlllP basic dpfinit.iolls and rpslIHs frolll [;l].

126 Mason

Definition. A binary relation .1 on a near-ring R is called a Boolean orthogonality if

H1 :1:.10 for all :r E R

B2 :1:.1y =? y.1:r

H3 :1:.1x =? :r = 0

H4 For all.T E R, ~: .1 = {yIJ:.1y} is an ideal of R

H5 .T.L.L n y.L.L = {OJ =? x.1y.

Note t.hat for all subs!'ts 5' f. ¢, ,S·.1 = [8].1

o rf- S, S.1 n oS' = {OJ if 0 E 8 .

{y.18 for all 8 E S}, and S.1 n S = ¢ if

Ideals of the form S.1 ,HI' cal\pd Jlolars and they form a colllpipte ortho-complplllentpd lattice

with n as inf and with sup giVPH by BvC = {BUC).L.1 = {8.LnC'.1).1. It. follows that for any idpals

.Ii , {L:.I;).1 = n.Ij.1 and (L:.!;).L.L = VN.1. Set D{R) = {[f').L.LIf' is a finit.p subsp!, of R}. This

sub-spmi-Ia.ttin' of pola.rs is a lattirp iff'v'x, y E R 3 ,dinit,e Bp!, FeR such tha.1. .'f,.L.L ny.L.L = [F].L.L.

Then D(R) is dif;tribntivp. This condition will figllfp prominpntiy in what. follows.

An ideal.! is railed a .1-idpal iffor all fillite subspts F' C.!, (f').1.1 C .!. An ideal.! is .1-primp

if wlH'nevpr A and Bare .1-ideals with An B <;; .! thpll A <;; .! or B <;; .!. Hpnrp if A B <;; An B for all

idpals A and B, primp ideals arp .1-prime. Equivalent.ly J is .1-prime iff :r.1.1 ny.1.1 <;; .! =? ;r E J or

y E J. WE' will writ!' (X) for thE' .1-ideal generatpd by X so (X) = U{[ F).L.LIF is finite , F' <;; )q. Finally when 1 :J .! are id!'als we say 1 is a divisor of .!.

Proposition 1.1 [3, Prop. 2.14] . Whpn D( R) is a Ia.ttin', earh .1-id!'al is !'It!' in!,prsertion of its .1-

prime .1-ideal divisors. Snch a divisor Q is a l1Iinimal .1-prime .1-ideal divisor of.! iff'v':r E Q 3y if- Q

such that x.1.1 n y.1.1 <;; .J. Moreover x.1y iff ;r.L.1 n y.1.1 = {O}.

If P is a .1-prime divisor of J define .I{ P) = {:I: E RI;I:.1.1 n y.1.1 <;; J for SOlllP Y if- Pl.

Proposition 1.2 [3, Prop. 2.1.5 a.nd 2.17). When D{ R) is a Ia.ttice

1. Each minimal.1-prime divisor of a. .1-ideal J is a .1-ideal (and so thp minilllal.1-prilllP divisors

of J are precisply t.hp minimal .1-prime .1-idpal divisors of .J).

2 . .I(P) is a .1-ideal and is the int!'rsertiou of all tltp lIlinilllal .1-primp divisors of ./ which are

conta.ined in 1'.

Mason 127

Of special note in this context ar!' thp ~-idpals 0(1') = {.rl:"~Y for SOlllp .II If. P}. Clearly

.1'1. <:;; P =} .1' If. O(P).

Cornish defines t.he a.lgphra.ic dosUfp systPIll J to 1)(' semiprilllp if thpfe is a lIlultipli('ation

defined on the IIlemlwrs of :1 such that

Sl A,B,C E J and A <:;; B =} AC <:;; BC' and CA <:;; CR.

S2 A2 = 0 =} A = 0

S3 AB <:;; A. n B for all A, BE J

S4 AB = {O} and AC = {O} =} A(B + C') = {O}.

Cornish remarked in [:~, p. S91] that sellliprilJ1e near-rings af(' pxalllpips. In vipw of S:~ and

S4, he Illust havl' had O-symnwt.ric ldt nl'ar-rings in mind. For right. IIl'ar-rilll!;S WP rpplnH' S4 hy

thl' correspolldiul!; right-hamled colldition. In that case all of [:~, Theorelll :1.'1] holds. In part.icular

we note that. .r~y {:} [x][y] = O. In t.he samp way a ff'du('ed zpro-sYIllIllPt.ric npar-rilll!; is reducpd in

the spnse of [3, p. SS:I] and then all of [;~, TIlPorPIll :1..5] holds. Thpli .1'~y {:} :1'.11 = O.

We now considpr the 3-sellliprillH' case. First we record t.hl' following rpsllit.

Lemma 1.3. The int.ersect.ion of a chain of :~-priIllP i<ipals is :~-priIllP. Evpry i(kal has minimal

3-prime divisors. In part.i('ular R has minimal :I-prilll(, id('a.iH.

Proof. Let I' = n Pi where the Pi a.re :\-prinw idea.ls and wp a.SSUllIP i ::; j =} Pi :::l Pi' If aRb C I'

and a If. P then a If. Pi for SOlllp i. Henc<' iJ E Pi a.nd so b E Pk for all k > i or plsp, sin('p aRb C Pk

thpn a E I'k C Pi. The sP('ond statelllPlit. follows from the obsprvat.ioll that. R itsplfis a :~-prilllp idpal

which ('an Iw llspd t.o hel!;ill n. dps(,Pllding chain of :~-prilllp i<ipals containing any idpall. Applying

this to J = (0) comptetps t.hp proof of HIP lemma.

Rpcall that. an idpall is fPfipxive (sep eg. [7]) if:t Ny <:;; 1 =} .II Rr <:;; J. We record ttl!' following

plpIllputary results from [I].

Proposition 1.4. [J, Theon'lll 2.11 and Lplllllla 2.10]. A :1-splllipriIllP i(iPal J IS rpfiexive a.nd

.TRy <:;; 1 =} [:r]R[y] <:;; T.

Theorem 1.5. If N is ;l-sPllJipriIllP, R ha.s a. Boolpan orthogolla.lity givPII by .r~y {:} :I'Ry = {O}.

128 Mason

Proof. Since R is 3-semiprime, it is semi prime so there is a Boolean orthogonality given by

xi.y ¢} [x][y] = {OJ. We show this is pquivalent. to xRy = {OJ. First suppose [:I:][Y] = {OJ. Then

for all primp ideals P either [x] C P or [y] C P, i.e. x E P or yEP. HPllCp :TRy C n P = {OJ.

Conversely if xRy = {OJ tilPn by Proposit.ioll lA, [.T]R[y] = {OJ C P for aU primp idpals P. Thus

either [x] C P or [y] C P so [:I:][Y] C n P = {OJ.

Remark. If R is a near-ring for which xRy = {OJ determinps an orthogonalit.y, HlP rpquirplllpnt

B2 forces R to be zero symmetric. For sineI' Or = 0 for all r, thPfpforp OR.T = {OJ for all :r a.nd

xRO = {OJ by B2. Hpnce .TOO = 0, i.p. xO = O.

Remark 2. When R is semiprimp, wp shall assUllle it has t.he ort.hogonality givell in Theorem 1 .. 5.

Lemma 1.6. If R is 3-semipri\Ile and if S is an ideal and l' a suospt. of R wit.h S n 1'.1 = {O}, then

S <;;; TH.

Proof. For all .5 E S, l/J E 1'.1, since Sand 1'.1 are ideals, we hav\' .sRw <;;; S n 1'.1

S <;;; TH.

Theorem 1. 7. If R is :l-semiprime then x.1.1 n y.1.1 = (.TRy ).1.1.

{OJ, i.p.

Proof. Since .T.1 <;;; (.TRy).1, .T.1.1 :2 (xRy).1.1 and so by symmetry :r.1.1 n y.1.1 :2 (J:Ry).1.1. To

prove thp wnvprsp supposp first t.hat ;; E x.1.1 n y.1.1 n (:rRy).1. Then;; E (:rRy).1 => ;;R.TRy = {OJ

so ;;R.T <;;; y.1. But since;; E y.1.1, ;;Rl: <;;; y.1.1 so ;;R;I; <;;; y.1 n y.1.1 = {OJ. Thus;; E :r.1. But also

;; E .T.1.1 so Z = O. Thus x.1.1 n y.1.1 n (:rRy).1 = {O} so by Lpl11ma 1.6, .7:.1.1 n y.1.1 <;;; (.TRy).1.1 as

rpquired.

Corollary 1.8. Every 3-prime ideal P is a i.-prime ideal.

Proof. :r.1.1 n y.1.1 <;;; P => (xRy).1.1 C P => :rRy <;;; P => ;r E P or yEP.

Proposition 1.9. When /l is 3-sPlIliprimp, tlll'n (a) ;r.1 is :1-splIliprim(' and (h) if P is a non­

minimal3-prime idea.l then p.1 = {OJ.

Proof. (a.) a/la C .T.1 => :c/laUa = {OJ so by 82, (lRxRa = {O} and nH.T!in/l.T = {O}. Hpnrp by

B3, aR.T = {OJ, and so (! E ;r.1.

(b) Suppose Q c P, Q primp. LI'\. (! E p.1, (! -1 O. Thf'n (! rt /-' so (! rt Q. But. a E p.1 => I/,Up = {OJ Vp E P and 3p E P\Q. TlwIl I/, E Q whirh is a cont.radict.ion.

Mason 129

Definition [3]. A m'ar-ring with Boolean orthogona.Jity .i is Rairl to have finiteness ronrlition "F"

if V.r, y E R 3 a finite set F <;; ;r-L-L n y-L-L such that ;c-L-L n y-L-L = F-L-L.

Theorem 1.10. Let R he a 3-semiprime near-ring.

The following are equivalent:

(a) R has condition "F".

(h) D(R) is a lattin' and each .i-prime .i-ideal is a :l-prime ideal.

Moreovf'1' undpr tlH'se ronditions

1. Every .i-ideal is an interspction of :l-prime ideals.

2. The spt of minimal .l-prilne .i-ideals coincides with the oet of minimal :l-prime ideals.

3. A 3-priuH' ideal P is minimal iff Vx E P 3y rf P sHrh that .r Ry = {O}.

4. For every 3-priuH' idpal P, O( P) is the intersection of all the minimal :l-prillle ideals contained

in P.

Proof. When (a) holds, D(R) is dearly a lattice. Lpt Q llP a. .i-primp .i-ideal surh that ;tRy <;; Q

for some x and y rf Q. By TheOfpm 1.7 1. her<' exist (11,"" rln E ;fRy s1Irh that ;t-L-L n y-L-L =

[(11, ... , (In]-L-L <;; Q. Sinre Q is .i-primf', x E Q or y E Q whirh is a contradiction.

Conversely suppose (h) holds. Let ;C,.IJ E R and put. .] = (xRy) = U{[F]UIF is finite,

F <;; ;/: Ry}. Then.] <;; (x Ny )-L-L = ;r-L-L n y-L-L hy TheOfPIn 1.7. S1IPpos!' .J c ;r-L-L n y-L-L. Thpn hy

Proposition 1.1 tllPre is a .i-prime .i-ideal Q containing.] fOf whirh ;r-L-L n y-L-L 1; Q. HPllre.r rf Q

and y rf Q, so ;rRy 1; Q. But ;r Ry c .J c Q which is a rontradiction. Therefore .J = ;r-L-L n y-L-L.

Since D( R) is a lattirp, .J = F-L-L for some finitp F as requirpd.

1. This follows from Proposition 1.1 and property "F".

2. If P is a minimal .i-primp .i-idpal it io :I-prime sinu' ;fHy C P => ;r-L-L n y-L-L = F-L-L for

some finite F C .rRy and F-L-L C P llPcause P is a .i-ideal. !!(,II('l'.r E P or yEP. If Q is

a 3-prilllP ideal with Q C P then Q io a .l-prillle idea'! hy Corollary IX By Proposition 1.1,

V;r E P 3y rf P such that ;r-L-L n y-L-L <;; Q. If WI' choos(' ;r E P\Q, thplI y E Q <;; P which

is a contradiction. Conversely if P is a minimal :1-prilllp idea.l it is a .i-primE' ideal. If Q is

a minimal .i-prime ideal mntaiupd in P. hy Proposition 1.2 it is iI -L-i<h'al so hy "F" it is

:l-primp and hener efl11als P.

130 Mason

:l. By (2) and Proposition 1.2, P is a minima.! :~-prilIle idea.! iff O( P) == P iff V:r E P 3y rf- P

with xLi n yLi == {OJ, i.p. (:rR.IJ).l.l == {O} hy Theorpm 1.7, i.p. :cRy == {O}.

4. This follows directly from (2) and Proposition 1.2.

Examples of near-rings with condition "F" an' reduced near-rings (whpf(' F == {:ry}) and 3-

semiprime near-rings with asn~nding chain condition on ideals. These examples have til!' additional

property that every reduced (n'spectively :l-semiprinlP) factor near-ring inlwrits property "F".

Of particular interpst in this regard is (I) of the prl'violls TllPoTPm sinn" as lllentionl'd in thl'

introduction, it remains an opl'n qul'stion as to precisely which conditions on R guarantf'f' that

any (or f'V!'TY) 3-sPllliprinH' ideal is an intersection of ;~-prilll(, i<il'a.!s. Sinn' (0) is a ..l-idpal in a

:~-semiprillle Il!'ar-ring we have

Corollary 1.11. If R is a :3-seTlliprillle lH'ar-ring with "F" then pvpry :l-spmiprimp ..l-idpal is an

intprsertion of :3-prillle idpa.ls. III particular (0) is such an intersection. Moreover pvpry 3-sPlllipriulP

idpal I ror which RI I has "F" is an intersection or :i-primp ideals.

Then' is another condition on R which kads to thf' conclusion that pVPry :l-semiprilllP idl'al is

a,n intersl'ction of :3-prilllp icIpals. This has he('n obtain!'d indppl'ndent.ly by Booth and Grol'nf'wald

[2]llsing a slightly diffpf!'nt ddinitioll of an "m-systpllI" bllt othPrwisp psspntiaHy thl' samp proof.

Dpfinp 1\1 C R to hI' a :hn-systpm if (/., b EM=} n1"b E IH for SOIl1P 1" E R. Also (IPfin(' R to havp

the USP condition if the union of ('vpry chain of :3-s(,lllipriIllP ideals is :1-sPllliprilllP. I am indehted

to Javed Ahsan for first suggpsting thp liSP of the idpal (1': Rn) in tl](' proof of the following.

Theorem 1.12. If q is a :1-sellliprilllP idpal ill a npar-ring with USP and q n M == <p for Romp

:lm-systpm M then thl'f{' pxist maximal elements ill S == p-sl'miprimp ideals 111:2 q

and In M == <p} and thes!' ar!' :l-priIlH'.

Proof. The USP condition Plisures that w!' C<I,n apply Zorll'S Lemma to S. Lf'j P Iw a maximal

dplllent in S. If a rf- P th(,11 J == (P : Ra) is :1-sPllliprilllP for if II rf- ./ thpn bRa rt P so ti]('rp pxists

1" E R, .. E U such that IJra"b1"n rf- J> and hpllcP b(1'fI..,)b rf- (P : Hn). Sin('(' l' <;;: (P : Ra), hy thl'

maximality of P pither l' == ( P : Ra) or :3 Ii E (P : Ra) n M. ASSIl1I1P thp lattpy is tYlIP.

Case 1. (/. EM. Sinn' a and II arp hoth in M, bra EM for sOllle 1". BlIt bE (P: Ra) =} bRa C P

which (,()lItraciicts tlH' fact that P n AI == <p.

Case 2. II rf- M. Sinn' bRa C P tllPn IIRb C P hy Proposition 1.4 so a E (P : Ilb). Applying the

Mason 131

argument used above for "a" to the element b EM, we see by Case 1 that P = (P : Rb). Hence

a E P which is again a contradiction.

Therefore we have P = (P: Ra) for all art P. Then if x, y E R\P, x rt (P : Ry) so ."CRy rt P.

Thus P is 3-prime.

Theorem 1.13. If R is 3-semiprime with USP then every 3-semiprime ideal of R is an intersection

of 3-prime ideals.

Proof. Let 1 be 3-semiprime. Clearly [~n{PIP is 3-prime and P;2 l}. Suppose Xu rt 1. Since

xoRxo rt 1 :lrl such that XI = xorlxo rt 1. Continuing we construct M = {xo,;Cj,:C2 ... } where

Xi E Xi-lRxi-l for i > 1 and M n 1 = cp. M is a 3m-system for if '''Ci, Xj, E R with j > i then

Xj = XiSXi for some S so Xj+J = Xjrj+JXj = (XisXirj)Xj = xitXj E M. By Theorem 1.12 there is a

3-prime ideal P with P n M = cp and P :J 1. Hence Xo rt P and we are done.

The USP condition has been mentioned for rings in [4] where it was observed that examples

include those in which every product of finitely generated ideals is finitely generated. These are

also examples of rings with "F" in the semiprime case. As to the examples of near-rings with

"F" mentioned previously, those with ace have USP, and Theorems 1.12 and 1.13 can be modified

to apply to reduced near-rings by dropping the USP requirement and replacing ":3-semiprime" by

"completely semi prime" . It is unknown whether "F" and USP are independent conditions.

In [11, p. 81] the authors define an ideal 1 of a reduced unital ring to be normal if (in our

notation) 1 = 1.1.1 and they show [11, Theorem 3.11] that 1 is normal iff 1 is an intersection of

the minimal prime ideals containing it. We show this is true for (not necessarily unital) semiprime

near-rings with "F" (and by anology for 3-semiprime near-rings with "F", if in the next two results

"minimal prime" is replaced by "minimaI3-prime"). First note that since Il.1 = {O} then for every

(minimal) prime ideal Q either 1 C Q or 1.1 C Q. In fact we have:

Lemma 1.14. If Q is a minimal prime ideal in any near-ring for which O( Q) = Q then for all

;r E R, either ;r E Q or x.1 ~ Q but not both.

Proof. If x.1 ~ Q then ."C rt O( Q) so ."C rt Q. Conversely.7: E Q => :ly rt Q such that Y E x.1 so

."C.1 rt Q.

N ow if 1 = 1.l.l then 1.1 -I (()) so there is some minimal prime ideal Q such that 1.1 rt Q. By

the above remarks, I C Q, i.e. every normal ideal is contained in sOllle minimal prime ideal.

132 Mason

Theorem 1.15. I is normal iff I = n{ PI P is a minimal prinw and P ::J f}.

Proof. In genera.l IH c:;; n{PIP i. r.1} c:;; n{ 1'11';;2 I}. Conversel y suppose

x E n{PIP ;;2 I} and y E r.1. WI' show [x][y] = {O} to show .T E rH. Givpn any minima.l primp

ideal Q, either I C Q or /.1 C Q, i.p. pit.hpr :r E Q or y E Q. Hpncp [:r][y] C nQ = {O}. Thm;

if I is normal I = [H == n{pIP;2 l}. Conversply I c:;; lH c:;; n{l'IP ;2 I} as notpd a hovp so if

[== n{PIP;2 I} then I = [H.

2. NORMAL NEAR-RINGS AND 1.-LOCAL IDEALS.

In this spetion WP will assume R is a zero-symmptric unital n('ar-rin~ and th at 1. is any

Boolean orthogonality on R .

Theorem 2.1. If R is a npar-ring wit.h Boolean ort.hogon a lity t.lWIl for all i<ip<l]s r,

1= n(O(M;) + I) wherp the intersection is over the set of ma.xima.l idp<l]s of R.

Proof. Suppos(' ;r. E D(Mi) + I for all ma.ximal iell'al, AIi . Thlls for p3.("h i, ;r = Zi + !Ji

where Zi E O( Mi), Yi E I . Thus Zi 1.wi for somelDi rt M i. By Proposit.ioll 1.1 zf.1 n 1I'f.1 =

{O}. Since R is zero symmetri c [Wi]Zi c:;; ",f.1 zf.1 c:;; 11>1-.1 n zf.1 == {O}. Sinn' {II';} is COIl-n

tained in no maximal ideal,R = I:[1Il;] so 1= L It; for som(' Il; E [Ill;] ([10, Theorelll 2.1]). I

Hence :r= (~Il;).T=1tl( ZI+Y!l+n2(:r2+Y2)+ ... +l/,n(zn+yn). Sinn' lI;Z; = 0 for all i.

Ui( Zi + Yi) = Ui(Zi + Yi ) -1/.iZi E [y,] c I . Hencp :r E I and WP havp shown t.hat. n(O(AJi ) + I) <;;: T.

The other inclusion is trivial.

Definition. A near-ring R with Boolean orthogonality 1. is said t.o he 1.-normal if :r1.y ~ .r.1+ y.1 =

R. This is a generalization of a definition which was giv(,1l for comlllutat.ive rings in [6] and llSprl in

both [8] and [9] for ("prtain semiprinw rings (and ha.s not.hing t.o do wit.h thp notat.ion of a norma]

ideal from the previous spdion). The next result.s and definit.iolls gpn eralizp thosp result.s in three

ways: They apply to any Boolean orthogonality in nPM-rings, ami nonp excpp!. Theorelll 2.7 rpquirf'

condition "F". Rat.her WP only assume for now that D( R) is a lil.ttin'.

Theorem 2.2. Th!' following a.ff' equiva\pnt

1. R is 1.-normal.

2. For all dist.inct minimal1.-primp idpals P and Q, P + Q = H.

3. EVNY propef prime idea.l conta.ins a IlUiql1P minimal1.-prillle i<!paI.

Mason 133

4. Every proppr maximal ideal contains a unique minimal l..-primp idpa.l.

5. For all prime ideals P, O( P) is a l..-prillle ideaL

6. For all maxima.l ideals M, O(M) is a l..-prinw ideaL

Proof. (I):::} (2). Given I' -:f Q, rnillimal1..-prilllP ideals, lpt 7' E I'\Q. TIH'n 7' E I' == 0(1') by

Proposition 1.2 so p1..$ for SOllie .~ rf- P. Hpllcf'.5.1 C P. Similarly p rf- Q :::} p.1 C Q. Thpreforp

R == s.1 + p.1 ~ P + Q.

(2) :::} (:l). A primp ideal P is l..-prime becanse if A, Bare l..-ideals with An 13 ~ P then

AB cAn B so A c P or B C P. Hpnce each prime ideal P cont.ains llIinilllal l..-prime ideals. If

P contains Q\ -:f Q2 then P:J Q\ + Q2 == R which is a clear rontradirtion.

(3) :::} (4) is trivial.

(4) :::} (5). Since O(P) is the intf'fspct.ioll of all tlw minimal l..-prilllP ideaJs contained in P,

and P ~ M for sam I' maxima.l idpa.1 M, it follows frolll (4) that. O(P) is l..-prilllp.

(5) :::} (6) is triviaL

(6) :::} (1). Suppose ;c1..y and .'1'.1 + y.1 C R so .'1'.1 + y.1 c AI for some maximal ideal M.

Bpcanse .'1' 1.. y, ;c.1.1 n y.1.1 == {O} ~ O(M) so;r E O(M) or y E O(M). But sinel' ;c.1 C M and

y.1 C M, .'1' rf- O(M) and y rf- O(M). This complptps thp proof.

Proposition 2.3. Whpn R is rpduced, R is l.-normal iff eitlH'r of thl' following conditions hold:

(7) (1'.5).1 == 1'.1 + .~.1 . 11. n

(R)IT1';==O:::}R==L>t. \ 1

Proof. Referring to the numbPrpd condtions in Thf'orl'lII 2.2 WI' see (X) :::} (I) sincp ;c1..y ¢} .1:Y == 0

when R is feducpd.

(I):::} (7) (1'8).1:J 1'.1 +8.1 sinn' if1'l E 1'.1,81 E .,.1 thpl1 (1"')(1'1 +8Jl == 1'81'1 +1'881 == 0+0

since a red uced llear-ri ng has IFP. Conversply [pI, t E (1'8).1. '1'1](>11 11' E 8.1 so hy ( I ) H == (11').1 + 8.1 .

Hence there exist .'1' E (11').1. Y E 8.1 wit.h 1 == ;r + y. Thpn I == ;c/ + .1/1 and ;rt E 1'.1. yl E 8.1 a.s

rp</uirpd.

134 Mason

(7) =? (8). By indudion (7) =? (~ri)1. =z=rf. T11I'n ITri=O=? (~)1. = R so

n

R = z= rf· I

Definitions.

1. An ideal I is railed pseudo l.-prime iff n1.1. n b1.1.

ni.b =? a E I or bEl).

n

{O} =? a E r or b E r (equiva.\ently

2. An ideal I is railed l.-Ioral if n[x - :ri]1.1. = {O} and Xi E r Vi =? :r E I. I

Lemma 2.4. a) A divisor of a i.-prime ideal is pseudo i.-prime and i.-local. An intersection of

i.-Ioral ideals is i.-Ioca.!.

Proof. a) If P C I for some ideal T and i.-prime ideal P and if 0,1.1. n b1.1. = {O} then a E P or

b E P, i.e. nET or II E I . ilenC(' I is pspudo l.-prime. Now P is l.-primp iff whenevpr Bi are 7~ ?A

l.-ideals with n Bi then Bi C P. Therefore if PeT a.nd n[:r. - Xi]1.1. = {O} wit.h :ri E r Vi then 1 I

[X - :/: ;]1.1. C Per for SOIl1P i so x - :ri E I, i.e . . r E 1.

n

b) If 1= n-h where the -h are l.-Iocal ideals and n[:r - :r;] = {O} with :ri E I Vi thPn ;/.'i E ·h Vi, k

so X E -h Vk, i.e. J: E I.

Lemma 2.5. If I is pseudo i.-primp a.nd is conta.ined in a, i.-primp idpal I' th('n 0(1') C I.

Proof. r E O(P) =? ri.s for somp 8 f. P whence 8 f. T. TllPreforp rEI.

Proposition 2.6. If R is l.-normal t.he pseudo l.-prime idl'als an' prl'ris('ly thl' divisors of i.-primp

ideals.

Proof. If I is a pSl'lHlo l.-primp ideal, reM for some maximal idpa,l M. Maximal idl'als arl'

primp, hencp i.-primp so O( M) c I hy Lemma 2.!i. Morl'ovpr O( M) is i.-primp Whl'll R is norma\.

The result now follows from Lemma. 2.4.

Theorem 2.7. If R is l.-normal pvery ideal is au int.ers!'ct.ioll of pSPIHlo l.-primp idl'aJs.

Proof. By Theorelll 2.1, I = n(O(M) + J) aud by Theorem 2.2 O(M) is l. -prinw so O(M) + I is

a divisor of a l.-primp idpal, i.f'. pSf'udo l.-prim!'.

Mason 135

Theorem 2.8. W1H'1l H has finiteness condition "F" an ideal I is .1-local iff I is an intersection

of divisors of .1-primp ideals.

Proof. By Lplllma 2.'1 an intprsprtioll of divisors of .1-prime idpals is .1-Iocal. Conversely let I be a

.1"local ideal and SUpfHlSP :r rf. I. Let S == {.1 - ideals .!I.! does not contain any intPrsertioll of the n

form n[J: - xJu with :ri E l}. TllPlI (0) E S or else.r E T since T is .1-local If {.J;} is a. chain of

idpals in S let J == U .Ii. CPftainly J is a .1-ideaL If J :J n[:r - :ri]J.J. with :ri E J, hy property "F" 1

3/11 ... bn such that.l:J [11[ .. . bn]J.J.. TIH'Tefore J :J {b l ... bn} so -h:J {111" .bn} for SOHlf' k. Since

·h is a .1-idpal ·h :2 [b l ... bn]J.J. == n[:r - .r;]J.J. which wntradicts that .h E S. Hence S is inductive

and by Zorn's Lpl1lma it has maximal elelllPnts. Let 1'1)(' OlIP. WI' show l' is .1-prime. Suppose,

therpforp, that (l,J.J. n IIJ.J. <;; P with (I, rf. P, Ii rf. P. The .1-ideals (1', (1,) and (I', b) lIlust ('ach wntaill "

an intersection of th!' form n[:r - :r;]1.1. and hPllCp so d(ws fl' == (P, n)n (P, Ii). We show that P :J fl', [

which contradicts thp fact that PES. If:r E II"\P, :r E [(1,,/,,12 .. '/I]J.J. n [b,g1" .g .. ]J.J. whpTe

f;, gi E I' Vi, j. U si ng thp distri hll tivity of JJ( R) and condition "F" we have :r E (aJ.J. nbJ.J. ) V pJ.J.

wllPrp F is finite and F C 1'. Sinn' (/,J.J. n bJ.J. c P hy assumption, :r E P whirh is tllP required

conclusion.

Corollary. If R is a .1-norma.1 [war-ring with finitelless condition "F" thpII ewry ideal is .1-Iocal.

REFERENCES

I. G. Birkellmeier, II. IIpatllPrly and E. Lpe, Prime ideals in Iwar-rings, Res. in Math. 24 (1993),

27-4X.

2. G. Booth and N. Groenpwald, v-Prime and 1,"Semiprinlf' Near-rings, slIhmittpd.

:l. W. Cornish, Boolean orthogonalitips and minimal primp idea,]s, Comlll. Algpllfa:l (I !l7!i),

X!i9-!lOO.

4 .. 1. Fishpf and R. Snidpr, On thl' VOli Nf'lIll1ann rl'glliarity of rings with I"Pglllar primp factof

rings, Pacific.l. Math. !i4 (197,1), l:l!i-1,14.

!i. N. Gropnl'waJd, Diff'pJ'('lIt primp ideals in n(',u'-rings, COIlLIII. Algphra 19 (I!l!ll), 2()(i7-267!i.

6. C. R. Illlijsmans atLd B. dp Pagtf'[", Idpal t1H'Ofy in f-algpllfas, Trans. AnH'r. Math. Soc. 269

( 19X2), 22!i-21!i.

136 Mason

7. G. Mason, Reflexive ideals, Comlll. Algebra 9 (1981), 1709-1724.

8. G. Mason and R. Raphael, A propos des id6aux locaux; rorrigendull\ et addendulll., Ann.

Sci. Math. Quebec 12 (1988), 255-261.

9. M. Parmenter and S. Stewart, Normal rings and local ideals, Math. Srand. 60 (1987),5-8.

10. G. Pilz, Near-rings (2nd Ed.), North-Holland, Amsterdam, 198a.

11. N. K. Thakare and S. K. Nimbhorkar, Space of minimal prime ideals of a ring without

nilpotent elelllPnts, .T. Pure Appl. Alg. 27, (19in), 75-85.

Department of Mathematics &, Statistics University of New Brunswick Fredericton, N. B. E3B 5A:l, Canada e-mail: GMASON@UNB.CA

Eingegangen am 15. September 1995