The abstract-algebra tag has no wiki summary.

**1**

vote

**2**answers

309 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

611 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

177 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

123 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

140 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

981 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 ...

**4**

votes

**1**answer

207 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 ...

**6**

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

118 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 ...

**0**

votes

**0**answers

113 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

106 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

367 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

509 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

152 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

275 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

292 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

310 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

256 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**

vote

**0**answers

280 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

301 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 ...

**3**

votes

**1**answer

465 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

424 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

186 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

158 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

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.
...

**0**

votes

**1**answer

419 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

310 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

142 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 ...

**3**

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 ...

**4**

votes

**3**answers

487 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

116 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 $Xâˆ’sâˆ’$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**

votes

**1**answer

269 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

374 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

291 views

### submonoids of Z_n

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

**0**

votes

**1**answer

263 views

### Group of divisibility

We know that the necessary condition for any partially ordered group to be a group of divisibility is that the group must be a directed group. What is the sufficient condtion for partially ordered ...

**1**

vote

**1**answer

365 views

### Integrally closed

Is there any idea to prove that $F + xk[[x]]$ is not integrally closed when the field $k$ is a proper extension of the field $F.$

**-1**

votes

**1**answer

771 views

### 1895 Math Trip problem on primitive roots of unity

How to prove that if $\theta _1,\theta _2,\theta _3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, ...

**1**

vote

**1**answer

167 views

### Ring of a Spectral Space

It is said, as far as I can tell that an arbitrary spectral space, i.e. a space that is $T_0$, sober and quasi-compact whose collection of quasi-compact open sets forms a basis and is closed under ...

**9**

votes

**1**answer

341 views

### Given a rational number a/b does there exist a finite group G and an automorphism f s.t. f maps exactly a/b elements of G to their own inverses?

I was helping a friend prepare for his intro abstract final and he mentioned the professor had once asked the question name a group and an automorphism that takes 3/4 of the elements of the group to ...

**0**

votes

**1**answer

384 views

### Weak algebraic structures

The following question can be thought as a sequel of this one.
Here I'm looking for a big list of example of weak algebraic structures: here weak means that the structure (i.e. operations) need not ...

**2**

votes

**1**answer

505 views

### Dedekind Spectra

Is there a class of ring spectra that corresponds to and/or extends the class of Dedekind rings from traditional algebra? Is there a notion of "ring of integers" of a ring spectrum? Additionally, is ...

**1**

vote

**0**answers

797 views

### Textbook suggestion for advanced algebra? [closed]

After having a solid year long undergraduate course in abstract algebra, I'm interested in learning algebra at a more advanced level, especially in the context of category theory.
I've done some ...

**5**

votes

**0**answers

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. ...

**7**

votes

**2**answers

691 views

### Virtual algebraic calculation within proofs

It seems to me that the undergraduates I teach have particular difficulty with proofs that involve reasoning about algebraic calculations that arise only theoretically. Since I have in mind doing ...

**2**

votes

**0**answers

169 views

### Idea behind choosing $\small f(x)$ as $c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$ in the proof that $\pi$ is transcendental.

I am going through the article at this link, where the author proves that: "$\pi$ is $\text{transcendental}$ over $\mathbb{Q}$". Although, I understand the proof, I have some doubts.
At page $6$, ...

**3**

votes

**2**answers

372 views

### Double orthogonal complement of a finite module

Crossposted from math.stackexchange since I'm not getting any answer.
Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the ...

**9**

votes

**2**answers

883 views

### Example of a Group which has $\text{SL}_{n}(\mathbb{Z})$ as the automorphism group

For the past one week, I have been trying to learn more about Automorphism groups of different groups. Very recently one of my friend asked this question to me:
What is the Autmorphism group of ...

**1**

vote

**1**answer

240 views

### Defect groups and subgroups

I have asked this question on Stack Exchange but had no response; it's been bugging me for a few days. I am struggling to see how to apply Mackey's theorem to prove a certain Lemma in Local ...

**5**

votes

**2**answers

1k views

### Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic.

Can anyone give me an example of:
An infinite abelian but non-cyclic group whose automorphism group is cyclic.