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5
votes
1answer
429 views

Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split? If not, is there an example? ...
2
votes
2answers
424 views

Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down?

Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in ...
1
vote
0answers
382 views

Kunneth spectral sequence

In Rotman's Homological Algebra, 1st edition, there is written: Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also ...
-3
votes
1answer
107 views

SHPS and SPHS inequality using monounary algebra

Let $A_n = \{(1,\ldots,n) , f \}$ where $f(i) = (i+1)$ if $i \neq n $ otherwise $f(n) = 1$. This describes a mono unary algebra. The proof for $HPS \neq SPHS$ I know uses metabelian groups and was ...
3
votes
2answers
386 views

Divisible torsion Z-modules

I am trying to prove that for any divisible torsion $\mathbb{Z}$-module $V$, this map $f:\mathbb{Q}/\mathbb{Z}\otimes\text{Hom}(\mathbb{Q}/\mathbb{Z},V)\longrightarrow V$ is an isomorphism via ...
1
vote
0answers
88 views

Positivstellensatz for non-polynomial term

Can we use Positivstellensatz (P-satz) below for a non-polynomial term? P-satz: Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the ...
3
votes
1answer
248 views

operations on ideals in a subring of number field

For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$, does this equality hold in general? $(I+J) \cap K = (I \cap K) + (J \cap K)$ I have no counterexample yet but I couldn't prove ...
6
votes
2answers
374 views

Misunderstanding in the hypotheses of Schlessinger's criterion

Good day to everyone. In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion. Let's fix a representation $\bar{\rho}$ of a group ...
-1
votes
1answer
158 views

Algebra generated by a tree [Edit] [closed]

Suppose that $(T,\leq)$ is a partially ordered set, we say $T$ is a tree* if for every $i\in T$, $\{s: s\in T, s\leq t\}$ is a well-founded chain. What I need to know is: Can the algebra ...
10
votes
0answers
198 views

Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...
3
votes
1answer
126 views

intersection growth of free profinite groups

Let $F_n$ be a free profinite group of finite rank $n$ and let $V_k$ denote the intersection of all open subgroups of $F_n$ of rank at most $k$ ($k \in \mathbb{N}$). My questions are: can I ...
2
votes
0answers
118 views

Hall's paper on the profinite groups and Andre Weils “voisinage” notion

I am reading through a classical paper A Topology for Free Groups and Related Groups by Marshall Hall Jr. in which profinite groups are defined for the first time. There he defines on p. 129: ...
0
votes
0answers
47 views

Decomposition results for locally commutative semigroups

Every finite abelian group is the direct product of its cyclic groups of prime order, and every commutative monoid divides a product of its cyclic submonoids. Could these results generalized to ...
-2
votes
1answer
178 views

finite index, self-normalizing subgroup of $F_2$ [closed]

Denote $F_2=\langle a, b\rangle$ to be the free group on two generators $a, b$. Let $H\leq F_2$ to be a subgroup with finite index $n$, so $H\cong F_{n+1}$ by Nielsen–Schreier theorem, recall that ...
9
votes
1answer
402 views

How is a descent datum the same as a comodule structure?

For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong ...
2
votes
0answers
154 views

Abelian Subgroup in an infinite non-abelian 3-group

Does an infinite non-abelian 3-group of exponent greater than or equal to 9 has an infinite abelian subgroup? I know that 2-groups and 3-group of exponent 3 has an infinite abelian subgroup. I wonder ...
3
votes
0answers
151 views

What can we say about a semigroup that's acted on by an ideal of polynomials?

This got no response on MSE, so posting here. Let $S = \{ (a,b) \in \Bbb{Z}^2 : \gcd(a,b) \neq 1 \} \cup (1,1)$. Then $S$ forms a semigroup. The operation being componentwise multiplication. Let ...
3
votes
1answer
475 views

Wedderburn decomposition of $D_{5}$

This is crossposted from MSE. The question: Find the Wedderburn decomposition of $D_{5},$ the dihedral group of order 10, over the field $\mathbb{F}_{3}.$ I have shown that the irreducible ...
2
votes
1answer
115 views

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 ...
2
votes
1answer
90 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 ...
5
votes
1answer
229 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 ...
0
votes
2answers
153 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 ...
6
votes
4answers
377 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 ...
8
votes
1answer
366 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 ...
2
votes
1answer
623 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 ...
5
votes
1answer
250 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 ...
1
vote
1answer
150 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 ...
1
vote
0answers
144 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 - ...
3
votes
2answers
525 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 ...
12
votes
2answers
377 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
0answers
98 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
1answer
101 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
1answer
169 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
0answers
101 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
1answer
177 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
0answers
76 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!
2
votes
0answers
119 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
1answer
836 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 $F$ ...
6
votes
1answer
440 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
0answers
92 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 ...
0
votes
2answers
332 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
0answers
114 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
3answers
532 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 ...
0
votes
0answers
105 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
2answers
397 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
1answer
685 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
1answer
189 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
1answer
133 views

Presentation of Semigroup

Is there a common presentation (set of equations) of the semigroup of functions on a given set(finite) ?
-1
votes
1answer
145 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
2answers
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 ...