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

Cleanliness in ringsMotivation

Group RingsResults

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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

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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

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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.

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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.

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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.

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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])

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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.

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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.

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Every left (right) Artinian ring is clean.

[A Artinian =⇒ A semiperfect]

Z is not clean.

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Z is not clean.

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Z is not clean.

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Z is not clean.

Polynomial rings are not clean.

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

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Z is not clean.

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Z is not clean.

A clean ring =⇒ Mn(A) clean ring.

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

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Z is not clean.

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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.

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

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Every homomorphic image of a clean ring is again clean.

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

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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]

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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.

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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.

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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.

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Question:

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Question:

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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?

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

Vas’s question:

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von Neumann algebras are ∗-rings;

it is simpler to work withprojections than with idempotents.

Vas’s question:

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Vas’s question:

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Vas’s question:

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Vas’s question:

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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.

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A = T2(Z2)

(a b0 c

)7→(

c b0 a

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A = T2(Z2)

A is a clean ∗-ring:

(a b0 c

)7→(

c b0 a

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A = T2(Z2)

(a b0 c

)7→(

c b0 a

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A = T2(Z2)

(a b0 c

)7→(

c b0 a

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A = T2(Z2)

(a b0 c

)7→(

c b0 a

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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 ).

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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

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G group;

∑g∈G αgg

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G group;

∑g∈G αgg

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G group;

∑g∈G αgg

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G group;

∑g∈G αgg

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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

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G group;

∑g∈G αgg

(classical involution in RG ).

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G group;

∑g∈G αgg

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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.

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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:

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Useful results

Proof:

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Proof:

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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.

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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.

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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.

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Useful results

Proof:

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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.

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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).

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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−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 ).

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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

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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

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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

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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

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∑nr−1i=0 (ωg)i ∈ FG ,

e2 = e, but e∗ 6= e.

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∑nr−1i=0 (ωg)i ∈ FG , e2 = e,

but e∗ 6= e.

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

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

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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].

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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).

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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).

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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].

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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].

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Local commutative rings and non-abelian groups

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

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.

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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.

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of order 2pk , with p a prime.

If p ∈ J (R), then RD2pk is ∗-clean

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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.

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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.

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QD2n is ∗-clean.

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QD2n is ∗-clean.

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QD2n is ∗-clean.

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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).

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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.

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

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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 ·...

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