# 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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### 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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### 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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### 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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### 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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### 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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### 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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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+... 0answers 131 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 ... 0answers 631 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 ... 1answer 332 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. ... 2answers 279 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" component is a ... 1answer 360 views ### 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 ... 0answers 119 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 ... 0answers 38 views ### 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 ... 1answer 272 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 ... 0answers 85 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 ... 1answer 123 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 ... 2answers 451 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}... 0answers 127 views ### 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. 2answers 170 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_{... 2answers 396 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 ... 0answers 179 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 ... 0answers 374 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 ... 2answers 773 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 \;|\; ...
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 $\mathbb{K}[H]$ ...