The abstract-algebra tag has no wiki summary.

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### Does this solution guarantee $det(A)=0$ where $A\in M(R)$? [closed]

Suppose $R$ is a commutative ring with identity $1$ and the following matrix equation holds:
$\begin{pmatrix} a_n & & \\
\vdots & \ddots & \\
a_1 ...

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votes

**1**answer

76 views

### Varieties of rational languages and (pseudo-)varieties of finite monoids, question regarding closure property

Let $\mathcal Rat(A)$ denote the class of rational (or regular) languages over the alphabet $A$, a subset $\mathcal V(A) \subseteq \mathcal Rat(A)$ is called a variety of (rational) languages iff
...

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**1**answer

198 views

### Polynomiality of functions over residue rings

Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a ...

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**2**answers

134 views

### sublattice generated by lattice points intersecting a convex set

Suppose that $M\subseteq \mathbb{Z}^n$ is a module such that $\mathbb{Z}^n/M$ is free and $S\subseteq \mathbb{R}^n$ is a bounded, symmetric (around $0$) convex set. Let $M'$ be the module generated by ...

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**4**answers

366 views

### How universal is operadic approach to studying algebras?

I have just started to read about operads, so this question might be silly.
So it seems to me that any "reasonable" class of algebras can actually be defined as a class of all algebras over a certain ...

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votes

**1**answer

345 views

### Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...

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votes

**1**answer

515 views

### Commutative associative rational binary operations

What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)
Feel free to re-tag if you can think of ...

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**1**answer

223 views

### growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$?

I am interested in the growth rate of this type of group: $G=\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$, where $\sigma(a)=\begin{pmatrix}x&y\\z&w\end{pmatrix}\in SL_2(\mathbb{Z})$, where $a$ is ...

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**1**answer

147 views

### If $F(a_1,\ldots,a_k)=0$ whenever $a_1,\ldots,a_k$ are integers such that $f(x)=x^k-a_1x^{k-1}-\cdots-a_k$ is irreducible, then $F\equiv0$

I'm trying to understand a proof of the following theorem (from section II of Hall's paper An Isomorphism Between Linear Recurring Sequences and Algebraic Rings):
If $F(a_1, \ldots, a_k)$ is a ...

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134 views

### Solving a system of rational functions

Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...

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votes

**2**answers

496 views

### Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...

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**1**answer

329 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 ...

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**0**answers

94 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 ...

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**1**answer

98 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 ...

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**1**answer

160 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

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100 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 ...

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**1**answer

173 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 ...

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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!

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114 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 ...

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**1**answer

655 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 $F$ ...

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**1**answer

424 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 ...

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83 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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315 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 ...

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**0**answers

106 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 ...

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437 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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104 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 ...

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352 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 ...

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**1**answer

640 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 $m$ ...

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181 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 ...

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**1**answer

127 views

### Presentation of Semigroup

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

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143 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.
...

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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 ...

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**1**answer

222 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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5k 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 ...

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**0**answers

122 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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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 ...

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**1**answer

113 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

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**1**answer

372 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 ...

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**1**answer

575 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 ...

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**1**answer

156 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 ...

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**1**answer

322 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 ...

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**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 ...

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**2**answers

319 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 ...

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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 ...

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296 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 ...

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319 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 ...

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**1**answer

541 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 ...

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432 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?

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**1**answer

194 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 ...

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162 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 ...