1
vote
0answers
41 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
151 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
189 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
289 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
143 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 ...
9
votes
2answers
443 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
243 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 ...
17
votes
3answers
763 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
101 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 ...
5
votes
2answers
344 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
337 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 ...
17
votes
4answers
909 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
142 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
139 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
310 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
310 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
501 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
429 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
34 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
147 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
423 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
252 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
918 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
183 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 $x\mapsto\overline{x}$ ...
21
votes
5answers
1k 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 ...
4
votes
1answer
351 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
503 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
228 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
488 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
506 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 ...
31
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
268 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
264 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
333 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
981 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
540 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
252 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 $ba=0$? ...
8
votes
3answers
1k 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
221 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
485 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
742 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
753 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
960 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 ...
5
votes
0answers
274 views

Is there an idempotent measure on the free LD system?

This is a follow up question to MO question "Idempotent measures on the free binary system?". Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law: ...
2
votes
0answers
254 views

What happens geometrically when you take associated-graded (or complete, …) of a group ring at its augmentation ideal?

I am interested in the following functor from Monoids (in $\text{Set}$) to Graded Lie Algebras (over a fixed field of characteristic $0$). (By "graded" I mean only that my Lie algebras have some ...
11
votes
3answers
857 views

Is there a 'nice' interpretation of virtual representations?

Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those ...
4
votes
0answers
257 views

Construction of an algebra with prescribed representation of the automorphism group.

For this discussion, $G$ is a compact semisimple Lie Group. For many of the common representations of compact groups, there is a realization of the representation as the automorphisms of some ...
12
votes
7answers
1k views

Superfluous definitions

It is well known that the axioms of a ring R with unity 1 imply that the underlying group must be commutative. For if a and b are elements of R, and writing + for the group operation then applying ...
1
vote
0answers
216 views

Modern books about orders and algebras on trees

Please help to find books about orders and algebras on trees. If there is no modern books, please advice good old ones! I'm more interested in finite trees (my current problem), but infinite ones are ...
6
votes
3answers
830 views

Does “finitely presented” mean “always finitely presented”, considered in general

I'm wondering about the question, "If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?" I know this is true for groups and for ...