9
votes
0answers
88 views

Detecting invertible elements in group rings by their images for finite quotients of the group

Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent). Consider an element ...
5
votes
0answers
185 views

A generalization of real characters on a group

Yesterday I understood that I can't live without this construction: Let $G$ be a group, $A$ an associative algebra over $\mathbb R$ and $n\in{\mathbb N}$. We consider a sequence of maps ...
1
vote
0answers
189 views

Kaplansky Idempotent conjecture and Extension theory

We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is the problem of nontrivial idempotents for group algebra $\mathbb{C}\Gamma$ where $\Gamma$ is a torsion free ...
3
votes
1answer
66 views

Intersection of Maximal Left Ideals with Finite Dimensional Quotient

Let $\Gamma$ be a finitely generated group and let $A=\mathbb{C}[\Gamma]$ be the corresponding group algebra over $\mathbb{C}$. Let $X$ be the set of all maximal left ideals of $A$ and let $X_0=\{I ...
16
votes
0answers
231 views

Kaplansky's unit conjecture and unique products

There are three conjectures on group rings that bear the name of Kaplansky (see for example this question). The 'unit conjecture' in the title of the present question is the strongest of them, and ...
1
vote
1answer
105 views

Modules “projective in a subcategory”

In my research I have come up with the following notion which I would like to learn more about. It may be very naive. Let $R$ be a ring, $M$ an $R$-module and $S$ a class or $R$-modules closed under ...
14
votes
1answer
517 views

Does GL_n(Z) have a noetherian group ring?

Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL_n(Z)$ been considered in the literature? Motivation: a recent trend has been to study ...
1
vote
0answers
44 views

On generating Euler Square of index q, q-1 (where q is any prime power)

Can somebody help me in generating Euler Square of index q, q-1 (where q is any prime power). Also, kidly tell me if there is any code availeble for generating the stated Euler Square. The details of ...
4
votes
1answer
156 views

Potentially identity elements in an Abelian group

I didn't see this problem before. I motivated by the questions Is every commutative group structure underlying at least one (unitary, commutative) ring structure A basic question about rings ...
1
vote
1answer
217 views

A question in ring theory

Is there an example of two groups $G_{1}, G_{2}$ such that there are two non isomorphic ring $R_{1}$ and $R_{2}$ such that the additive group of both rings is isomorphic to $G_{1}$ and their unit ...
1
vote
2answers
304 views

A basic question about rings

Perhaps this is a trivial question, but I have no idea how to justify it. Call a pair of groups $(G_1, G_2)$ ring-compatible if $G_1$ is abelian and there exists a ring $R$ with addition and ...
2
votes
1answer
174 views

Monoids and groups of fractions

Let $G$ be a group containing a monoid $M$ that spans $G$ as a group. Is it possible to have a proper quotient $\varphi \colon G \to Q$ of $G$ such that the restriction of $\varphi$ to $M$ is ...
8
votes
2answers
464 views

Origin of the Socle of a module

Where does the notation $\mbox{Soc}(M)$ (the sum of all simple submodules of a module $M$) first appear?
4
votes
1answer
279 views

Rings with group of units cyclic of prime order

For what prime numbers $p$ there exists a ring with identity and exactly $p$ invertible elements ? REMARK It can be shown that for $p=5$ there is no such ring, so I am wondering for what values of ...
18
votes
3answers
831 views

Why is $(\mathbb{Z}/3\mathbb{Z})^3$ not a class group of an imaginary quadratic number field ?

In general, it seems not known which finite abelian groups are class groups of quadratic number fields. For imaginary quadratic number fileds, I read that $(\mathbb{Z}/3\mathbb{Z})^3$ is the smallest ...
2
votes
1answer
109 views

Amenable group rings embeddable in skew fields

I've made this question on math.stackexchange.com (also offering a bounty) but I did not receive any answer: I'm looking for a reference of the following fact: given a (countable?) amenable group ...
6
votes
2answers
371 views

“Inverse problem” for Brauer groups

This question is just a curiosity, but I'm really interested in the answer. It was originally posted on math.stackexchange ...
2
votes
2answers
421 views

Is there an almost-direct product decomposition for disconnected reductive algebraic groups?

$\textbf{Some definitions:}$ Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...
19
votes
4answers
956 views

Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?

Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...
2
votes
0answers
156 views

Minimal prime ideals of a group ring

Let $R$ be a left Noetherian ring (if you prefer you can just think to $R$ as a skew field, I'll be happy with an answer under that hypothesis) and $G$ be a polycyclic-by-finite (or, if you prefer ...
2
votes
0answers
140 views

Simple groups analogous to fields

I recently realized for the first time (while teaching my undergraduate abstract algebra course) that simple groups and fields share the analogous property that neither has (nontrivial) factor ...
1
vote
2answers
345 views

Semiring naturally associated to any monoid?

For any monoid $M$, we can naturally construct a semiring $S$ as follows: Let the additive monoid of $S$ be the free commutative monoid on $M$ Let the multiplicative monoid of $S$ be $M$ Then, if ...
3
votes
1answer
322 views

A version of the group ring using direct product rather than direct sum?

Let $G$ be an infinite group. It's (integral) group ring $\mathbb{Z}[G]$ has as its elements the finite formal linear combinations $$ m_1g_1 + m_2g_2 + \cdots + m_ng_n,\qquad n\in\mathbb{N},\quad ...
12
votes
2answers
516 views

The set of orders of elements in a group

Let $A$ be a subset of natural numbers. Consider the following problem: Is there a group $G$ such that $\lbrace O(x) \; | \; x \in G \rbrace = A\cup\lbrace 1\rbrace$ ? (where $O(x)$ is the order of ...
7
votes
3answers
442 views

Why/when classification of simple objects is “simple” ? E.g. (unknown) classification of simple Lie algebras in char =2,3…

Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem. I wonder what is known/expected for char p=2,3 ? More vague ...
0
votes
0answers
39 views

How to distinguish projective involutions?

Let $K$ be a field of characteristic not $2$ and $R$ a continuous von Neumann regular ring with centre $Z=Z(R)$ isomorphic to $K$. For an example one may assume $R$ is a matrix ring of $n\times ...
1
vote
1answer
154 views

fixed point scheme in caracteristic p

Let X\rightarrow A^{n} a smooth affine scheme over an affine space. Everything is defined over a field k. Let G a finite group acting on X and suppose that his order is divisible by the caracteristic ...
4
votes
0answers
431 views

Elementary polynomial-free proofs of fundamental theorem of Galois theory?

I am looking for simple proofs that show the correspondence between intermediate fields in a field extension and subgroups of the Galois group. I'm happy for everything to be subfields of ...
2
votes
1answer
255 views

Reference request (or otherwise): Adjoint action

I am interested in clearing up a confusion of mine. I will try to make my question as clear as I can but I apologize in advance if this is not the case. Given a unitary group of some unital ...
6
votes
3answers
1k views

Why are ring actions much harder to find than group actions?

I admit freely that the following question is a bit of a fishing expedition inspired by this lovely "definition" of a module as found on Wikipedia: A module is a ring action on an abelian group. ...
4
votes
1answer
189 views

Heisenberg-type groups over rings with involution

Hello everyone! In a paper "Coverings of twisted Chevalley groups over commutative rings" Eiichi Abe intoroduced the following construction: Let $R$ be a commutative ring and $x\mapsto\overline{x}$ ...
22
votes
5answers
2k views

“Lie algebra” for a general group ?

Is there analog of Lie algebra for the case of topological groups which are not necessarily differentiable manifolds, and in particular for finite groups? here by "analog" i mean that it should have ...
5
votes
2answers
488 views

integral equivalence classes of quadratic forms

Let $A$ and $B$ two symmetric matrices definite positive over $\mathbb{R}$. Then we say that $A$ and $B$ are integrally equivalent if there exists $Q\in GL_n(\mathbb{Z})$ such that $A=Q.B.Q^t$ (1) ...
3
votes
1answer
554 views

Why do elementary matrices generate the special linear group over polynomial rings?

Let $R$ be a ring. An elementary matrix over $R$ is a matrix with $1$s along the diagonal and at most one other nonzero entry. Let $\text{EL}_n(R)$ denote the subgroup of $\text{GL}_n(R)$ generated by ...
1
vote
1answer
230 views

Restrictions of Modules and Dimensions

Let K be a finite field and let R,P be groups (with R a subgroup of P). I know that the irreducible KP-modules have dimensions 1,4 and 16 over K. I have a KP-module M, and I know that M has dimension ...
11
votes
2answers
522 views

inverse limits of group algebras and profinite groups

For an inverse system {$G_i$} of finite groups, and a fixed field $\mathbb{k}$, one can consider the corresponding group algebras $\mathbb{k}[G_i]$. The latter form an inverse system of ...
5
votes
2answers
520 views

When is $\mathbb{G}_m(R)$ enough to determine $R$?

Say I have a ring, $R$, with 1 which I consider my universe, and I know its group of units $G=\mathbb{G}_m(R)$. Then given a subgroup, $H\le G$, can I determine if there is there a subring $S_H$ such ...
35
votes
1answer
2k views

What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
5
votes
0answers
276 views

Can any group be realized as the multiplicative group of a ring? [duplicate]

Possible Duplicate: Ring with Z as its group of units? Given a group $G$, does there always exist a ring $R$ such that $R^\times \cong G$? I feel like this isn't true but that's just a hunch. ...
1
vote
0answers
265 views

Can we make a useful ring on an Elliptic curve? [closed]

I know that given an elliptic curve $E$ we can define an addition $+$ over the set of points on the curve to make in an abelian group. However Can we define multiplication on $E$ in a natural way so ...
5
votes
2answers
335 views

Hall's treatment of algebraic operations

Marshall Hall, in his famous book Theory of Groups, does not always require a binary operation be "well-defined", i.e. an operation is a relation instead of a function (there might be more than one ...
13
votes
1answer
1k views

Ring with Z as its group of units?

Is there a ring with $\mathbb{Z}$ as its group of units? More generally, does anyone know of a sufficient condition for a group to be the group of units for some ring?
8
votes
1answer
545 views

Automorphisms of a matrix in Smith normal form?

Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. These matrices are ...
3
votes
1answer
257 views

Group ring and left zero divisor II

Let $K$ be a finite field and $G$ be a discrete group. Is it true that for every $a=e+a_1+\ldots+a_n,b=e+b_1+\ldots+b_m\in K[G]$ with $b_i\neq e,a_j\neq e$ the condition $ab=0$ implies $ba=0$? ...
11
votes
3answers
2k views

Zero divisor conjecture for finite fields

I believe that the most attractive "zero-divisor" conjecture is the existence of non-trivial zero-divisors in a group ring $\mathbb{C}[G]$ for a torsion free group $G$. For the sake of knowledge let ...
9
votes
0answers
227 views

What do Multilinear Forms tell us about Representations?

The last few days I have been calculating whether certain group representations are real, complex, or quaternionic. It is well-known that the type of the representation corresponds to what type of ...
12
votes
1answer
491 views

Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?

Due to the first (and very helpful) answer I received, I've reformulated the question a little: $G$ and $H$ are now assumed to be $p$-groups. Let $\{p}$ be a prime, and let $\mathbb{F}_p$ be the ...
24
votes
1answer
806 views

Strong group ring isomorphisms

Background/Motivation Let $R$ be a commutative ring with unit. If $G$ is a finite (or in general, discrete) group, let $R[G]$ be the group $R$-algebra associated to $G$. The isomorphism problem for ...
12
votes
3answers
781 views

For which rings R is SL_n(R) generated by transvections?

Let $R$ be a commutative ring $R$ with $1$, and $n \geq 2$ an integer. Under which conditions is the group $SL_n(R)$ generated by transvections? (A transvection is a matrix with $1$ everywhere ...
10
votes
3answers
990 views

Group ring and left zero divisor.

Let $K$ be a finite field and $G$ be a discrete group. Is it true that for every $a,b\in K[G]$ the condition $ab=0$ implies $ba=0$? It does not seem to be related to zero divisor problem, any ...