cleanliness and group ringshomepages.vub.ac.be/~abachle/gryb/slides/veloso_slides.pdf ·...

Cleanliness in ringsMotivation

Group RingsResults

References

Cleanliness and group rings

Paula Murgel Veloso

Universidade Federal Fluminense (UFF), Niteroi – RJ, Brazil

June 21, 20171

1Groups, Rings and the Yang-Baxter EquationPaula Murgel Veloso Cleanliness and group rings


Group RingsResults

References

Clean and ∗-clean rings

A an associative ring with 1a ∈ A is a clean element if it may be written as a = u + e, forsome unit u ∈ A and some idempotent e ∈ A.

a ∈ A, with A a ∗-ring, is a ∗-clean element if it may be written asa = u + p, for some unit u ∈ A and some projection p ∈ A.

Examples and properties

Any field is clean and ∗-clean.

Any Boolean ring is clean.

Every local ring is clean and ∗-clean.

Paula Murgel Veloso Cleanliness and group rings


Group RingsResults

References


A an associative ring with 1

a ∈ A is a clean element if it may be written as a = u + e, forsome unit u ∈ A and some idempotent e ∈ A.








Group RingsResults

References


A an associative ring with 1a ∈ A is a clean element if it may be written as a = u + e,

forsome unit u ∈ A and some idempotent e ∈ A.








Group RingsResults

References










Group RingsResults

References



a ∈ A, with A a ∗-ring, is a ∗-clean element if it may be written asa = u + p,

for some unit u ∈ A and some projection p ∈ A.







Group RingsResults

References










Group RingsResults

References

Clean and ∗-clean rings (contd.)

Examples and properties (contd.)

Every semiperfect ring is clean.

[A is a semiperfect ring if A/J (A) is semisimple andidempotents lift modulo J (A).][A is semiperfect ⇐⇒ A is clean and has no infinite set oforthogonal idempotents]

Every left (right) Artinian ring is clean.[A Artinian =⇒ A semiperfect]

Z is not clean.

Polynomial rings are not clean. (Nicholson & Zhou, 2004 [10,Proposition 13])

A clean ring =⇒ Mn(A) clean ring. (J. Han & Nicholson,2001 [4, Corollary 1])



Group RingsResults

References



Every semiperfect ring is clean.[A is a semiperfect ring if A/J (A) is semisimple andidempotents lift modulo J (A).]

[A is semiperfect ⇐⇒ A is clean and has no infinite set oforthogonal idempotents]


Z is not clean.





Group RingsResults

References



Every semiperfect ring is clean.[A is a semiperfect ring if A/J (A) is semisimple andidempotents lift modulo J (A).][A is semiperfect ⇐⇒ A is clean and has no infinite set oforthogonal idempotents]


Z is not clean.





Group RingsResults

References




Every left (right) Artinian ring is clean.

[A Artinian =⇒ A semiperfect]

Z is not clean.





Group RingsResults

References





Z is not clean.





Group RingsResults

References





Z is not clean.

Polynomial rings are not clean.

(Nicholson & Zhou, 2004 [10,Proposition 13])




Group RingsResults

References





Z is not clean.





Group RingsResults

References





Z is not clean.


A clean ring =⇒ Mn(A) clean ring.

(J. Han & Nicholson,2001 [4, Corollary 1])



Group RingsResults

References





Z is not clean.





Group RingsResults

References

Clean rings ∗-clean rings (cont.)


ΠiAi clean ring ⇐⇒ each Ai clean ring.

(Nicholson & Zhou,2004 [10, Example 3])

Every homomorphic image of a clean ring is again clean.(Nicholson & Zhou, 2004 [10, Theorem 22])

A subring of a clean ring may not be clean.

Every ∗-clean ring is a ∗-ring and is clean.



Group RingsResults

References



ΠiAi clean ring ⇐⇒ each Ai clean ring. (Nicholson & Zhou,2004 [10, Example 3])






Group RingsResults

References




Every homomorphic image of a clean ring is again clean.

(Nicholson & Zhou, 2004 [10, Theorem 22])





Group RingsResults

References









Group RingsResults

References

Historical background

Nicholson defined clean rings in 1977 [9] while studying exchangerings (= suitable rings).[A ring is said to be exchange if idempotents can be lifted moduloevery left ideal]If all idempotents are central, then every element in an exchangering is the sum of a unit and an idempotent.

J. Han and Nicholson were the first to tackle cleanliness in grouprings in 2001. [4]



Group RingsResults

References


Nicholson defined clean rings in 1977 [9] while studying exchangerings (= suitable rings).

[A ring is said to be exchange if idempotents can be lifted moduloevery left ideal]If all idempotents are central, then every element in an exchangering is the sum of a unit and an idempotent.




Group RingsResults

References


Nicholson defined clean rings in 1977 [9] while studying exchangerings (= suitable rings).[A ring is said to be exchange if idempotents can be lifted moduloevery left ideal]

If all idempotents are central, then every element in an exchangering is the sum of a unit and an idempotent.




Group RingsResults

References


Nicholson defined clean rings in 1977 [9] while studying exchangerings (= suitable rings).[A ring is said to be exchange if idempotents can be lifted moduloevery left ideal]If all idempotents are central, then every element in an exchangering is the sum of a unit and an idempotent.




Group RingsResults

References

Historical background (contd.)

Notice: clean rings are the “additive analogues” of unit-regularrings.

Question:

What exactly is the relationship between unit-regular rings andclean rings?

Theorem (Camillo & Khurana, 2001 [1, Theorem 1])

A ring A is unit-regular ⇐⇒ for every a ∈ A, there is u ∈ U(A)and an idempotent e ∈ A such that a = e + u (i.e., A is a cleanring) and aA ∩ eA = 0.



Group RingsResults

References


T. Y. Lam’ question:

(at the Conference on Algebra and Its Applications, 2005, OhioUniversity)

Which von Neumann algebras are clean as rings?

von Neumann algebras are ∗-rings; it is simpler to work withprojections than with idempotents.

In 2010, Vas proposed the definition of ∗-clean rings in [11].

Vas’s question:

Are there clean ∗-rings that are not ∗-clean?



Group RingsResults

References



(at the Conference on Algebra and Its Applications, 2005, OhioUniversity)Which von Neumann algebras are clean as rings?



Vas’s question:




Group RingsResults

References




von Neumann algebras are ∗-rings;

it is simpler to work withprojections than with idempotents.


Vas’s question:




Group RingsResults

References






Vas’s question:




Group RingsResults

References


Example (C. Li & Zhou, 2011 [6, Example 2.6])

A = T2(Z2)

A is a clean ∗-ring: ∗ :

(a b0 c

)7→(

c b0 a

);

for every involution ∗, A is not ∗-clean.



Group RingsResults

References



A = T2(Z2)

A is a clean ∗-ring:

∗ :

(a b0 c

)7→(

c b0 a

);




Group RingsResults

References



A = T2(Z2)

A is a clean ∗-ring: ∗ :

(a b0 c

)7→(

c b0 a

);




Group RingsResults

References

Involutions in group rings

C. Li & Zhou [6] were the first ones to tackle ∗-cleanliness in grouprings in 2011. However, very little is known about the subject evennow.

G group;

classical involution in G : g 7→ g−1.

If R is a commutative ring, the R-linear extension of the groupinvolution in G yields a ring involution in RG : ∗ : RG −→ RG ,α∗ =

∑g∈G αgg

−1 (classical involution in RG ).



Group RingsResults

References


C. Li & Zhou [6] were the first ones to tackle ∗-cleanliness in grouprings in 2011.

However, very little is known about the subject evennow.

G group;



∑g∈G αgg




Group RingsResults

References



G group;



∑g∈G αgg




Group RingsResults

References



G group;


If R is a commutative ring, the R-linear extension of the groupinvolution in G yields a ring involution in RG :

∗ : RG −→ RG ,α∗ =

∑g∈G αgg




Group RingsResults

References



G group;



∑g∈G αgg

−1

(classical involution in RG ).



Group RingsResults

References



G group;



∑g∈G αgg




Group RingsResults

References

Abelian caseNon-abelian case

Useful results

Theorem (Nicholson, 1972 [8])

Let R be a ring and G be a group. If R is local, G is a locallyfinite p-group and p ∈ J (R), then RG is a local ring.

Theorem (C. Li & Zhou, 2011 [6, Theorem 2.2])

A commutative ∗-ring A is ∗-clean ⇐⇒ A is clean and everyidempotent in A is a projection.

Proof:

[⇒] A is clearly clean. Let e2 = e ∈ A. Since R is ∗-clean, we havee = p + u, where p ∈ A is a projection and u ∈ U(A). So,u = e − p e u(e + p − 1) = (e − p)(e + p − 1) = 0; thene = 1− p. Therefore, e∗ = 1∗ − p∗ = 1− p = e.



Group RingsResults

References


Useful results


Let R be a ring and G be a group.

If R is local, G is a locallyfinite p-group and p ∈ J (R), then RG is a local ring.



Proof:




Group RingsResults

References


Useful results





Proof:




Group RingsResults

References


Useful results





Proof:

[⇒] A is clearly clean.

Let e2 = e ∈ A. Since R is ∗-clean, we havee = p + u, where p ∈ A is a projection and u ∈ U(A). So,u = e − p e u(e + p − 1) = (e − p)(e + p − 1) = 0; thene = 1− p. Therefore, e∗ = 1∗ − p∗ = 1− p = e.



Group RingsResults

References


Useful results





Proof:

[⇒] A is clearly clean. Let e2 = e ∈ A. Since R is ∗-clean, we havee = p + u, where p ∈ A is a projection and u ∈ U(A).

So,u = e − p e u(e + p − 1) = (e − p)(e + p − 1) = 0; thene = 1− p. Therefore, e∗ = 1∗ − p∗ = 1− p = e.



Group RingsResults

References


Useful results





Proof:

[⇒] A is clearly clean. Let e2 = e ∈ A. Since R is ∗-clean, we havee = p + u, where p ∈ A is a projection and u ∈ U(A). So,u = e − p e u(e + p − 1) = (e − p)(e + p − 1) = 0; thene = 1− p.

Therefore, e∗ = 1∗ − p∗ = 1− p = e.



Group RingsResults

References


Useful results





Proof:




Group RingsResults

References


Commutative local rings and cyclic groups

Theorem (Gao, Chen & Y. Li, 2015 [2])

Let R be a local commutative ring and C3 = 〈g〉 the cyclic groupof order 3.

If 3 ∈ J (R), then RC3 is ∗-clean.

If 3 6∈ J (R), then RC3 is ∗-clean ⇐⇒ RC3 is clean and theequation X 2 + X + 1 = 0 has no solutions in R.



Group RingsResults

References


Local commutative rings and cyclic groups (contd.)

Theorem (Y. Li, Parmenter & Yuan, 2015 [7])

Let R be a local commutative ring.

If n ∈ J (R), with 3 ≤ n ≤ 5, then RCn is local and ∗-clean.

If n 6∈ J (R), with 3 ≤ n ≤ 5, then RCn is ∗-clean ⇐⇒ RCn isclean and Φn(X ) = 0 has no solutions in R/J (R).

If 3 ∈ J (R), then RC6 is ∗-clean ⇐⇒ RC6 is clean.

If 2 ∈ J (R) or 6 6∈ J (R), then RC6 is ∗-clean ⇐⇒ RC6 isclean and Φ3(X ) = 0 has no solutions in R/J (R).



Group RingsResults

References


Fields and abelian groups

Let F be a finite field with char(F) = p > 0 and G be a finiteabelian group of exponent nr , with p - nr . (D. Han & Ren, 2017[3])Consider ω ∈ F an nthr primitive root of 1.Then FG is not ∗-clean: let g ∈ G and |〈g〉| = nr ,e = 1

nr

∑nr−1i=0 (ωg)i ∈ FG , e2 = e, but e∗ 6= e.

Theorem [3, Theorem 1.1]

Consider ω 6∈ F an nthr primitive root of 1.Then FG is ∗-clean ⇐⇒ ∃t such that pt ≡ −1 (mod nr ).



Group RingsResults

References



Let F be a finite field with char(F) = p > 0 and G be a finiteabelian group of exponent nr , with p - nr . (D. Han & Ren, 2017[3])

Consider ω ∈ F an nthr primitive root of 1.Then FG is not ∗-clean: let g ∈ G and |〈g〉| = nr ,e = 1

nr






Group RingsResults

References



Let F be a finite field with char(F) = p > 0 and G be a finiteabelian group of exponent nr , with p - nr . (D. Han & Ren, 2017[3])Consider ω ∈ F an nthr primitive root of 1.

Then FG is not ∗-clean: let g ∈ G and |〈g〉| = nr ,e = 1

nr






Group RingsResults

References



Let F be a finite field with char(F) = p > 0 and G be a finiteabelian group of exponent nr , with p - nr . (D. Han & Ren, 2017[3])Consider ω ∈ F an nthr primitive root of 1.Then FG is not ∗-clean:

let g ∈ G and |〈g〉| = nr ,e = 1

nr






Group RingsResults

References



Let F be a finite field with char(F) = p > 0 and G be a finiteabelian group of exponent nr , with p - nr . (D. Han & Ren, 2017[3])Consider ω ∈ F an nthr primitive root of 1.Then FG is not ∗-clean: let g ∈ G and |〈g〉| = nr ,

e = 1nr






Group RingsResults

References




nr

∑nr−1i=0 (ωg)i ∈ FG ,

e2 = e, but e∗ 6= e.





Group RingsResults

References




nr

∑nr−1i=0 (ωg)i ∈ FG , e2 = e,

but e∗ 6= e.





Group RingsResults

References




nr






Group RingsResults

References




nr



Consider ω 6∈ F an nthr primitive root of 1.

Then FG is ∗-clean ⇐⇒ ∃t such that pt ≡ −1 (mod nr ).



Group RingsResults

References




nr






Group RingsResults

References


Fields and abelian groups (cont.)

Theorem (D. Han & Ren, 2017 [3, Theorem 1.2])

Let F be a field with char(F) = 0 and G be a finite abelian groupof exponent nr . Consider ω 6∈ F an nthr primitive root of 1.Then FG is ∗-clean ⇐⇒ F(ω + ω−1) ( F(ω).


Let F be a field with char(F) = 0 and G be a finite abelian groupof exponent nr = pk or 2pk , with p an odd prime and k ∈ Z,k > 0. Consider ω 6∈ F an nthr primitive root of 1.Then FG is ∗-clean ⇐⇒ 2 | [F(ω) : F].



Group RingsResults

References




Let F be a field with char(F) = 0 and G be a finite abelian groupof exponent nr .

Consider ω 6∈ F an nthr primitive root of 1.Then FG is ∗-clean ⇐⇒ F(ω + ω−1) ( F(ω).





Group RingsResults

References




Let F be a field with char(F) = 0 and G be a finite abelian groupof exponent nr . Consider ω 6∈ F an nthr primitive root of 1.

Then FG is ∗-clean ⇐⇒ F(ω + ω−1) ( F(ω).





Group RingsResults

References









Group RingsResults

References






Let F be a field with char(F) = 0 and G be a finite abelian groupof exponent nr = pk or 2pk , with p an odd prime and k ∈ Z,k > 0.

Consider ω 6∈ F an nthr primitive root of 1.Then FG is ∗-clean ⇐⇒ 2 | [F(ω) : F].



Group RingsResults

References






Let F be a field with char(F) = 0 and G be a finite abelian groupof exponent nr = pk or 2pk , with p an odd prime and k ∈ Z,k > 0. Consider ω 6∈ F an nthr primitive root of 1.

Then FG is ∗-clean ⇐⇒ 2 | [F(ω) : F].



Group RingsResults

References









Group RingsResults

References


Local commutative rings and non-abelian groups


Let R be a local commutative ring with 2 ∈ J (R). Then RS3 isclean, but not ∗-clean.


Let R be a local commutative ring.

If 2 ∈ J (R), then RQ8 is ∗-clean.

If 2 ∈ U(R), then RQ8 is ∗-clean ⇐⇒ RQ8 is clean and theequation X 2 + Y 2 + Z 2 + 1 = 0 has no solutions in R.



Group RingsResults

References


Local commutative rings and dihedral groups

Theorem (Huang, Y. Li & Tang, 2016 [5])

Let R be a local commutative ring and D2pk be the dihedral group

of order 2pk , with p a prime. If p ∈ J (R), then RD2pk is ∗-clean


Let R be a local commutative ring with 2 ∈ J (R), and D2n thedihedral group of order 2n. If n is not a power of 2, then RD2n isnot ∗-clean.



Group RingsResults

References





of order 2pk , with p a prime.

If p ∈ J (R), then RD2pk is ∗-clean





Group RingsResults

References










Group RingsResults

References







Let R be a local commutative ring with 2 ∈ J (R), and D2n thedihedral group of order 2n.

If n is not a power of 2, then RD2n isnot ∗-clean.



Group RingsResults

References










Group RingsResults

References


Fields and dihedral/quaternion groups


QD2n is ∗-clean.


If gcd(q, 2n) = 1, then FD2n is ∗-clean.


If 4 | n and gcd(q, 2n) = 1, then FQ2n is not ∗-clean.



Group RingsResults

References

V. P. Camillo, D. Khurana, A characterization of unit-regularrings, Communications in Algebra 29 (2001), 2293 – 2295.

Y. Gao, J. Chen, Y. Li, Some ∗-clean Group Rings, AlgebraColloquium 22 (2015) 169–180.

D. Han, Y. Ren, On ∗-clean group rings over abelian groups,Journal of Algebra and Its Applications 16 (2017), 1750152(11 pages).

J. Han, W. K. Nicholson, Extensions of clean rings,Communications in Algebra 29 (2001), 2589–2595.

H. Huang, Y. Li, G. Tang, On ∗-clean non-commutative grouprings , Journal of Algebra and Its Applications 15 (2016)1650150 (17 pages).



Group RingsResults

References

C. Li, Y. Zhou, On strongly ∗-clean rings, J. Algebra Appl. 10(2011) 1363–1370.

Y. Li, M. M. Parmenter, P. Yuan, On ∗-clean group rings,Journal of Algebra and Its Applications 14 (2015).

W. K. Nicholson, Local group rings, Canadian MathematicalBulletin 15 (1972), 137 – 138.

W. K. Nicholson, Lifting idempotents and exchange rings,Transactions of the AMS 229 (1977), 269 – 278.

W. K. Nicholson, Y. Zhou, Rings in which elements areuniquely the sum of an idempotent and a unit, GlasgowMathematical Journal 46 (2004), 227 – 236.



Group RingsResults

References

L. Vas, ∗-Clean rings; some clean and almost clean Baer∗-rings and von Neumann algebras, J. Algebra 324 (2010),3388–3400.



Group RingsResults

References

Thank you for your attention!

(officially not celebrating Eric Jespers’ 62nd birthday)Paula Murgel Veloso Cleanliness and group rings

cleanliness and group ringshomepages.vub.ac.be/~abachle/gryb/slides/veloso_slides.pdf ·...

Documents