The abstract-algebra tag has no wiki summary.

**11**

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**1**answer

323 views

### $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.
Denote the ...

**2**

votes

**0**answers

92 views

### Lazard's $\Gamma_n(f)$ as cocycle

In Michel Lazard's "Commutative Formal Groups" Springer Lecture Notes, he defines an operator on a polynomial 3-cochain $f$ denoted $\Gamma_n(f)$, which defines as the $n^{th}$ homogeneous piece of ...

**1**

vote

**1**answer

94 views

### “Symmetric” Polynomial 4-cocycles

It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a ...

**2**

votes

**1**answer

154 views

### English translation of Steinitz 1910?

Does there exist an English translation of Steinitz' 1910 work "Algebraische Theorie der Körper"?
http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002167042

**2**

votes

**0**answers

98 views

### classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant

Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem ...

**0**

votes

**1**answer

169 views

### Find a special element in group algebra

Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the ...

**0**

votes

**0**answers

69 views

### Example of a ring whose minimals are annihilators of idempotents?

I'm looking for examples† of rings with the property that for each
$P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$)
† other than domains!

**1**

vote

**0**answers

111 views

### Is a certain group related to a primitive L function isomorphic to $Gal(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ for some $\ell$?

I define the notion of "Galois class of L functions" in the following way:
$A$ is a Galois class of L functions if and only if the follwing three conditions hold simultaneously:
1) every element ...

**3**

votes

**1**answer

574 views

### Diagonalize the simultaneous matrices and its background

For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:
Question1:Is there always a
nonsingular matrix $P$ over the same
field $F$ ...

**5**

votes

**1**answer

416 views

### What fields can be used for an inner product space?

Math people:
The title is the question: What fields can be used for an inner product space?
This question has been discussed in Math Stack Exchange with no definitive resolution. A similar question ...

**4**

votes

**0**answers

82 views

### How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...

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votes

**2**answers

306 views

### An exercise about Tor [closed]

Let $I$ and $J$ be two ideals in $A$. Show that
$\operatorname{Tor}_{1} (A/I, A/J) =\frac {I \cap J} { IJ} $
and
$Tor_{2} (A/I, A/J) =\ker(I \otimes_ {A}J \to IJ )$.
The first Tor is not a ...

**2**

votes

**0**answers

105 views

### Quivers for algebras which are not basic or unital.

Are there any definitions of quivers for algebras which are not basic or unital? I am reading the book Elements of the representation theory of associative algebras: volume one. The ordinary quivers ...

**2**

votes

**2**answers

405 views

### Stanley-Reisner ring of a simplicial complex is a functor?

Let $K$ bea field and $[n]=\{1,\ldots,n\}$ and $K[x]=K[x_1,\ldots,x_n]$. For $\sigma=\{i_1,\ldots,i_k\}\subseteq [n]$, denote $x_\sigma=x_{i_1}\cdots x_{i_k}=\prod_{i\in\sigma}x_i\in K[x]$. Let ...

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**0**answers

103 views

### Operating on a set of sequences - such as adding sequences and so on possible even when sequence is coded as number?

From http://math.stackexchange.com/questions/346680/operating-on-a-set-of-sequences-such-as-adding-sequences-and-so-on-possible-ev
Suppose that there is a way to code some set of sequences into ...

**1**

vote

**2**answers

338 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 ...

**12**

votes

**1**answer

630 views

### Number of idempotent $n\times n$ matrices over $\mathbb{Z}_m$ ?

Is there any known formula for the number of idempotent $n\times n$ matrices over $\mathbb{Z}_m$ ?
The number of idempotent matrices over a finite field is well-known and since we can decompose $m$ ...

**1**

vote

**1**answer

179 views

### jacobian polynomial

Here is the question which could be quit difficult (but could be not):
Let $C$ be a field of complex numbers and $f \in C[x,y]$ be a polynomial such that there exist
$g \in C[x,y]$ and $Jac(f,g) \in ...

**2**

votes

**1**answer

125 views

### Presentation of Semigroup

Is there a common presentation (set of equations) of the semigroup of functions on a given set(finite) ?

**-1**

votes

**1**answer

142 views

### product of two non-decomposible (closed) polynomials

Let $C$ be the field of complex numbers. Polynomial $f \in K[x,y]$ is called closed or non-decomposible if
$C[f]$ is algebraically closed (definition for $f \in K[x1,...,xn]$ is the same).
Theorem.
...

**5**

votes

**2**answers

1k views

### Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...

**5**

votes

**1**answer

215 views

### Hochschild homology and change of non-ground ring

Let $k$ be a field, $R$ is a commutative algebra over $k$ and $A$ is an associative algebra over $R$. There is a morphism of commutative algebras $R \to T$. Is it possible to reduce calculation of ...

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votes

**12**answers

4k views

### Useless math that became useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.
My idea is to amend my article with some theories that seemed useless when they are created but ...

**2**

votes

**0**answers

120 views

### Clifford algebra is graded separable

Let $D$ be an algebra of odd differential operators on a free module $V$, this algebra is isomorphic to the Clifford algebra $Cl(V^* \oplus V)$. Let $m$ denote multiplication map $$m : D\otimes D \to ...

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**0**answers

114 views

### Structure groups and a special class of L-functions

Hello,
Let $X$ and $Y$ be two mathematical objects such that there exists a canonical embedding $f:X\hookrightarrow Y$. I define the structure group of $Y$ relatively to $X$, denoted $Str(Y/X)$, as ...

**0**

votes

**1**answer

109 views

### composition factor of a group which isomorphic to the alternating group of order 7

I want to find groups whose composition factor is isomorphic to the alternating group of order 7, which groups have this condiction?
best regards

**6**

votes

**1**answer

371 views

### Two questions about commutative theories

Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook of Categorical ...

**1**

vote

**1**answer

551 views

### How to solve a system of equations over permutations?

Imagine you have a $n\times n$ matrix filled in with permutations over $n$ elements. Now you pick one permutation from each row randomly starting from the first row and by multiplying them get a ...

**4**

votes

**1**answer

154 views

### Detecting elements of nilpotent extensions via finitely generated ones

Suppose that $\pi:A \to K$ is a surjective map of $K$-algebras with nilpotent kernel, for $K$ a commutative unital ring field. (Recall that this means $Ker(\pi)^n=0$ for some integer $n$.) Notice that ...

**3**

votes

**1**answer

301 views

### Character theory of $2$-Frobenius groups.

This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...

**1**

vote

**1**answer

294 views

### Question about the shape of the dual module of a module

Hi,
I am reading this paper and have the following question:.
On page 5 you can see different types of $A_n$-modules ($\ A_n=k[x,y]/\langle x^2,y^{n+2},xy^{n+1}\rangle\ $)$\ $ and later in the paper ...

**3**

votes

**2**answers

315 views

### Guidelines to prove that 2^sqrt{2} is a transcendental number? [closed]

After a small search that I did I was unable to spot any answers here.What I am trying is to prove why the 2^sprt{2} is transcendental number.I know that this probably is a closed problem and probably ...

**1**

vote

**1**answer

259 views

### About some kind of “converse” of a theorem from Galois theory

Hello,
It is a well known result from Galois theory that, given a Galois extension $L$ of a field $K$, an element $x$ of $L$ is in $K$ if for all $\sigma$ in $Gal(L/K)$, one has $\sigma(x)=x$.
My ...

**1**

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**0**answers

286 views

### about union of conjugate proper subgroups in a math paper

My question is about the shaded area in this image. Does the symbol $L=\bigcup_{g \in G} T^{g}$ means that $L$ is a union of sets or $L=\langle T^{g}, g\in G \rangle$? If it means the first one, then ...

**4**

votes

**2**answers

311 views

### orbits of independent sets of the hypercube

How does one enumerate the distinct orbit classes of independent sets of the hypercube modulo symmetries of the hypercubes?
The counting of the number of independent sets in an n-dimensional ...

**4**

votes

**1**answer

508 views

### the direct sum of injective modules need not be injective

the Bass-Papp Theorem asserts that a commutative ring $R$ is Noetherian iff every direct sum of injective R-modules is injective. Thus every non-Noetherian ring carries a counterexample.
If
$$
I_1 ...

**5**

votes

**1**answer

428 views

### What is the origin of the term magma?

Wikipedia credits Bourbaki with coining it, but doesn't provide a source. Does anyone happen to know the motivation for using this term?

**0**

votes

**1**answer

192 views

### Is there a subfield $F$ of $\mathbb{R}$ such that $\mathbb{R}$ is a finite algebraic extension of $F$. [duplicate]

Possible Duplicate:
Examples of algebraic closures of finite index
The question is in the title.
I can prove that if such field $F$ exist then the extension $\mathbb{R}/F$ cannot be of ...

**1**

vote

**0**answers

161 views

### Invariant Ideals in Split Hopf Algebroids

Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following:
An ideal $J\subset S$ is invariant under the action of the group ...

**6**

votes

**3**answers

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.
...

**0**

votes

**1**answer

456 views

### Maximal subgroups of a finite p-group

I want to prove the following:
Let $G$ be a finite abelian $p$-group that is not cyclic.
Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup ...

**1**

vote

**2**answers

330 views

### Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?

Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that ...

**2**

votes

**1**answer

160 views

### Bimodule version of IBN

Hello all,
Does anyone have an example in mind of a ring $R$ for which $R^n\cong R^m$ as $R,R$ bimodules for some positive integers $n\neq m$?
I would be a little surprised if someone showed no such ...

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votes

**0**answers

105 views

### Making an algebra the uniqe maximal one-sided ideal in another unital algebra

If $R$ is an algebra without a unit, then the standard unitisation $R^\sharp$ can have maximal one-sided ideals other than $R$. Thus, it is natural to ask about the following. Let $R$ be an algebra ...

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votes

**3**answers

508 views

### Octic family with Galois group of order 1344?

Does the octic,
$\tag{1} x^8+3x^7-15x^6-29x^5+79x^4+61x^3+29x+16 = nx^2$
for any constant n have Galois group of order 1344? Its discriminant D is a perfect square,
$D = ...

**0**

votes

**0**answers

119 views

### On X-s-permutable subgroups of a finite group

I want to prove Lemma 2.1(1) in the paper On X-s-Permutable Subgroups of a Finite Group by Min Bang SU, Yang Ming LI. It is on the web.
This is my proof.
.
Since $H$ is $X−s−$permutable in $G$, then ...

**1**

vote

**1**answer

197 views

### why evry p group of infinity order is not simple?

why every p group of infinity order is not simple?

**2**

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**1**answer

273 views

### Noether Normalization in $\mathbb{C}[[x_1,…,x_n]]$

Hello everyone,
though I am quite sure my question is not on a research level, I am a bit helpless as of where to ask it. I posted it on math.stackexchange here, but there has been no answer helping ...

**5**

votes

**1**answer

393 views

### Left ideals vs right ideals

By default, let all algebras be complex and unital. I am concerned with the non-commutative algebras. I am wondering if the following might be true (at least for some classes of algebras, like ...

**1**

vote

**2**answers

293 views

### submonoids of Z_n

Anyone knows how to describe explicitly the submonoids of Z_n, regarded as a multiplicative
monoid?