Questions tagged [group-rings]

A group ring $R[G]$ is a ring constructed in a natural way from a ring $R$ and a group $G$.

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Product of Hermitian forms over a group ring

Let $G$ be a group and $k=Z[G]$ the corresponding group ring equipped with the standard involution. Let $(P,p)$ and $(Q,q)$ be two $\epsilon$-quadratic forms, $\epsilon = \pm1$, where $P$ and $Q$ are ...
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Nullstellensatz and nilpotence of a module

Let $\nu : G \rightarrow H$ be a surjective group homomomorphism with kernel $N$, $H$ abelian, and $G$ finitely generated. The rational abelianization of $N$, $H_1(N)$ is a $\mathbb{C}[H]$-module, ...
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4 votes
1 answer
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Idempotents in group rings of finite cyclic groups

For which fields $K$ and integers $n>1$ does the group ring $K(\mathbb{Z}/n\mathbb{Z})$ have idempotents distinct from $0$ and $1$?
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2 votes
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Group rings of free abelian groups

Is it true that for free abelian finitely generated groups $G_1$ and $G_2$, if $\mathbb Z[G_1]\simeq \mathbb Z[G_2]$, then $G_1\simeq G_2$? If yes, is there any reference to such a fact?
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0 votes
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zero divisors of group ring when the group is abelian

Let G be an abelian group with torsion and C[G] be the group ring over complex numbers C. Is there a clear description or classification of zero divisors of C[G]?
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When is the profinite completion of a Noetherian group ring also Noetherian?

Let $G$ be a group, and let $\mathbb{Z}[G]$ denote its group ring. Its profinite completion is the inverse limit over all ideals of finite index. By Benjamin Steinberg's answer here, this profinite ...
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Wedderburn decomposition of semisimple group algebras

Let $G$ be a finite $p$-group. What can we say about the Wedderburn decomposition of the group algebra $FG$? Here $F$ is a finite field of characteristic co-prime to $p$. Can we say something in the ...
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12 votes
1 answer
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Group ring with infinite stable rank

In searching for a counterexample in homological stability, I came across the following question: Is there a known example of a finitely presented group $G$, so that the group ring $\mathbb{Z}[G]$ ...
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9 votes
1 answer
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When is the augmentation ideal projective as RG-module?

Let $G$ be a finite group and let $R$ be a commutative ring. I'd like to ask, if there is a theorem of the following kind: The augmentation ideal $I_G$ is projective as RG-module, if and only if ... ?...
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9 votes
2 answers
189 views

Residual finiteness for modules over group rings

Let $G$ be a finitely generated residually finite group and let $M$ be a finitely generated $\mathbb{Z}[G]$-module. Question: Must $M$ be residually finite in the sense that for all nonzero $x \in M$, ...
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8 votes
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Units of group algebra of dihedral group

Question: Can we fully describe the group of units (=invertible elements) $(KG)^\times$ of the group algebra $KG$ for $K=\mathbf{F}_2$, $G=D_\infty=\langle s,t|s^2=t^2=1\rangle$, the infinite ...
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4 votes
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Is the group ring of an amenable group, viewed as multiplicative monoid, amenable?

Motivated by this question, it seems natural to ask the following: Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group $G$ so that the ...
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21 votes
4 answers
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Units in the group ring over fours group after Gardam

Giles Gardam recently found (arXiv link) that Kaplansky's unit conjecture fails on a virtually abelian torsion-free group, over the field $\mathbb{F}_2$. This conjecture asserted that if $\Gamma$ is a ...
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5 votes
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Integral monoid rings and Ore conditions

Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$. I have two, ...
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2 votes
0 answers
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Units in group rings

Let $F$ be any field with $p$ elements and $G$ be any finite $p$-group, combining together they form a group ring $FG$. And $V(FG)$ denotes group of units of coefficient-sum equal to 1 in $FG$. We ...
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6 votes
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How much does Ext tell me about isomorphisms?

So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...
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2 votes
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Dimension of center of $k[G]/\mathrm{rad}k[G]$ when characteristic of $k$ divides the order of $G$

Let $G$ be a finite group and consider $k[G]$ where $k$ is a field. In the scenario where $\mathrm{char}(k)$ divides $|G|$, how can one show that the dimension of $Z(k[G]/\operatorname{rad}k[G])$ is ...
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2 votes
0 answers
139 views

Flat augmentation ideal of a group-ring

If $G$ is a group and $I$ the augmentation ideal $I=Ker(\mathbb{Z}G\rightarrow \mathbb{Z})$ suppose that: $I$ is a flat (right) $\mathbb{Z}G$-module. $I$ is a finitely generated (right) $\mathbb{Z}G$...
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3 votes
1 answer
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augmentation ideal is always finitely generated?

$G$ is a finitely presented group (but not a finite group), and $\mathbb{Z}G$ is the corresponding group ring. $I$ is the kernel of the augmentation morphism $\mathbb{Z}G\rightarrow \mathbb{Z}$. Is $...
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4 votes
1 answer
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Groups for which all projections of $C^*_{\text{red}}G$ belong to $\mathbb{C}G$

Revision: According to comment of Wojowu we give a complete revise for this post. A group $G$ is a pr-group if all projections of $C^*_{\text{red}} G$ are contained in its dense subalgebra $\mathbb{...
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0 votes
1 answer
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A subset (or subgroup) associated to a group

Edit: According to comment conversations we revise the question. Let $G$ be a group. We consider the following subset of $G$: $$\{g\in G \mid e^{\lambda_g} \in \mathbb{C}\lambda (G)\},$$ where $\...
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8 votes
1 answer
668 views

A question regarding Kadison-Kaplansky idempotent conjecture (A nearest group element $g$ to a nontrivial self adjoint unitary element u )

Edit: According to answer and comments by Prof. Valette we edite the question. The Kadison Kaplansky conjecture says: Kadison-Kaplansky conjecture: If $G$ is a torsion-free discrete group then $C^*_{\...
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2 votes
0 answers
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Partially commutative elements in powers of augmentation ideal

Let $\vartheta$ a relation of parcial commutation over a set $X,$ and consider the respective free parcially commutative group $F(X, \vartheta).$ Let $K[F(X, \vartheta)]$ the parcially commutative ...
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8 votes
1 answer
488 views

Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions

(This question is originally from Math.SE, where it didn't receive any answers.) $\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\ext}{...
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2 votes
1 answer
246 views

Flatness of submodules of free modules

Are submodules of free $\mathbb{Z}[G]$-modules flat? if not what conditions on $G$ makes it true? $G$ is an infinite group. If $\mathbb{Z}[G]$ is a Prüfer domain then this is true. Can a group ring $...
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5 votes
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Any f.p. faithful simple module over a primitive group ring?

Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$. There ...
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10 votes
0 answers
312 views

Is the norm element characteristic in modular group rings?

Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}_pG$ as $\mathbb{F}_p$-algebras and let $n = \sum_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$? ...
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3 votes
0 answers
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Does this element belong to $\mathbb CG$?

Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...
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5 votes
0 answers
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Are these element in a group algebra of a torsion-free group zero divisors?

Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements‌ can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)? $$1+x+y,\quad 4+x+x^{-1}+y+...
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5 votes
0 answers
148 views

lifting of idempotents in group ring

Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In ...
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34 votes
0 answers
920 views

Groups whose complex irreducible representations are finite dimensional

By a complex irreducible representation of a group $G$, I mean a simple $\mathbb CG$-module. So my representations need not be unitary and we are working in the purely algebraic setting. It is easy ...
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14 votes
1 answer
510 views

Group rings such that every (countably generated) module has a maximal submodule

Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma. I am interested in the following question, with two variants. ...
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7 votes
2 answers
378 views

Using Dunwoody's results on cohomological dimension to learn about a von Neumann regular group ring

Just recently I've stumbled across Warren Dicks' book Groups, trees and projective modules (1980) and I was pretty stunned. I know nothing of group cohomology, but I gather the "tree" ...
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3 votes
1 answer
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Looking for example of quotient of group algebra by ideal of group ring which fails to be injective

I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is ...
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5 votes
0 answers
125 views

Chains of right annihilators in group rings

See the update below This problem emanates from a question on not-so-simple random walks on finitely generated groups. But to explain the connections would require an extremely long essay. Let $G$ be ...
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1 vote
0 answers
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When is genus the same as stable equivalence?

Suppose $M, N$ are two $R$-modules (I had in mind the group ring $R=\mathbb{Z}[G]$ for a finite group $G$). By localizing at a prime $p$ I mean $M_{(p)}\cong M\otimes_R R_{(p)}$. If $M$ and $N$ are ...
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6 votes
1 answer
342 views

Transcendence degree of the fraction field of $k[G]$ for torsion free abelian group $G$

Let $k$ be a field of characteristic $p$ and $G$ be a torsion free abelian group . Then the group ring $k[G]$ is an integral domain , let $k(G)$ denote its field of fractions . Then can we say ...
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2 votes
0 answers
87 views

Improved dimension subgroups

Given a group $G$, one can define (see below) a descending sequence of subgroups $K^s(G)$, $s=1,2,\dots$, satisfying $$ \gamma^s(G)\subseteq K^s(G)\subseteq D^s(G), $$ where $\gamma^s(G)$ is ...
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4 votes
1 answer
134 views

Right reversibility of submonoids of nilpotent groups

Let $G$ be a finitely generated group (optionally torsion-free). Let $N$ be a submonoid of $G$ (that is, a subsemigroup with $1$). A (cancellative) monoid/semigroup $S$ is right reversible if for ...
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6 votes
2 answers
465 views

In the group ring $\mathbb{Z}_p [G]$, what elements satisfy $(\sum a_g g)^p = \sum a_g g^p$?

Here $\mathbb{Z}_p$ is the ring of integers in $\mathbb{Q}_p$. Preferably I would want to know this for a general group $G$, but I have been concentrating on the case $G = (\mathbb{Z} / p^n \mathbb{Z}...
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Is every stably free module of commutative group ring free?

Is every stably free module of commutative group ring free? if not can you give me an example of commutative group ring with nonfree stably free module.
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6 votes
2 answers
200 views

groupring morphisms and bialgebra

Let $G_{1}$ and $G_{2}$ be two groups. Suppose that we have a morphism $\mathbb{Z}[G_{1}]\rightarrow \mathbb{Z}[G_{2}] $ of bialgebras is it true that this morphism comes from a morphism of groups $G_{...
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  • 1,543
1 vote
2 answers
498 views

Jacobson radical of group algebra

For a finite group G and a finite field $\mathbb{F}_p$ of characteristic $p$, J($\mathbb{F}_{p^k} \otimes_{\mathbb{F}_p }\mathbb{F}_p G ) = J(\mathbb{F}_{p^k}G)$? where $J(\mathbb{F}_{p^k}G)$ is the ...
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  • 11
7 votes
0 answers
288 views

The augmentation filtration on a group ring

Let $G$ be a group and $\mathbb QG=\{\sum a_ig_i\colon a_i\in\mathbb Q, g_i\in G\}$ its group ring over $\mathbb Q$. It is a Hopf algebra. Let $I=\{\sum a_ig_i\colon\sum a_i=0\}$ be the augmentation ...
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11 votes
0 answers
383 views

Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module $M = R / (ax + by + c) R$. I am ...
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13 votes
2 answers
794 views

Torsion-freeness of two groups with 2 generators and 3 relators and Kaplansky Zero Divisor Conjecture

Let $G_1$ and $G_2$ be the groups with the following presentations: $$G_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$ $$G_2=\langle a,b \;|\; ...
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1 vote
0 answers
639 views

Invertible elements in a group algebra

Let $H$ be a torsion-free abelian group and let $\mathbb{K}$ be a field with two elements. I would like to ask the following question: Is the group of units of the group algebra $\mathbb{K}[H]$ ...
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  • 1,543
1 vote
0 answers
182 views

The normalizer problem for group rings

I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = $\cal{U}$$(\...
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8 votes
2 answers
737 views

Must a finitely generated projective module over a group ring with vanishing coinvariants be trivial?

Let $G$ be a (possibly infinite) group. Let $\mathbb{Z}[G]$ be its integral group ring and let $P$ be a finitely generated projective module over $\mathbb{Z}[G]$. Suppose that the coinvariants of $P$ ...
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1 vote
1 answer
230 views

Why do we not lose any generality by proving it only for finitely generated groups [closed]

In the proof of following theorem, in a paper by Farkas- Here $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ and $U_1(\mathbb{Z}G) $ is the set of normalized units of the integral group ring $\...
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