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4
votes
0answers
121 views

Can we drop commutativity assumption?

Let $A$ be an associative algebra with a unit over a field $k$. fix $n > 1$. Define a $k$-algebra structure on the vector space $A^{\otimes n} = A \otimes_k \dots \otimes_k A$ (where there are $n$ ...
0
votes
2answers
105 views

Making idempotent element by a relation [on hold]

Let $R$ be a commutative ring with identity and let $a, b \in R$ such that $a=ab$. How can we make a non zero idempotent element of $R$ by this relation?
-2
votes
0answers
58 views

a question on maximal ideals in rings [closed]

If $R$ is a commutative ring with unit and $p$ is a prime number ($2,3,5,\cdots$), then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$? If $R$ is a field then it is not ...
3
votes
0answers
84 views

Cardinality based results in Topological Vector Spaces?

Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen: TVS's $V$ of ...
1
vote
0answers
127 views

An identity satisfied by “Differentiation”

I asked this question in MSE but I did not received any answer. So I repeat it here: Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order ...
-2
votes
0answers
56 views

Jacobson radical of an indecomposable commutative ring [migrated]

Let $R$ be a commutative indecomposable ring with identity which has infinit many maximal ideals. Can we deduce that $Jacobson$ radical, $J(R)$, (the intersection of all maximal ideals)of $R$ is th ...
-1
votes
0answers
21 views

application of modular theory on matrices [migrated]

i am very new to group theory .... just started reading infact. I could really appreciate some guidance on the following problem with which i am stuck. Let G be the set of all 2x2 matrices [a b c d] ...
0
votes
0answers
49 views

The trace ideal of a non zero $R$-module [migrated]

Let $R$ be a commutative ring with identity and $M$ be a cyclic $R$-module, we may define the ideal $tr(M)$ associated with $M$, the sum of the ideals $f(M)$, for all $R$-homomorphisms $f \in ...
3
votes
0answers
62 views

Symmetric tensor powers as tensors over symmetric group algebra [migrated]

Let $V$ be a $k$-vector space and $V^{\otimes n}$ the $n$-fold tensor power of $V$ and let $\mathbb{S}_n$ be the symmetric group of an n-element set, with its signum representation denoted by ...
0
votes
1answer
112 views

Is it possible to generalize a result of Wang?

Assume $A$ and $B$ are commutative algebras with $1$. There is a nice result of Wang, Corollary 8, which says the following: "Let $B = A[z] = A[Z]/(h(Z))$. Then $B$ is a separable algebra over $A$ if ...
0
votes
0answers
11 views

Finding minimal polynomials [migrated]

How would you find the minimal polynomial for $\zeta_5$ over $\mathbb{Q}(\cos(2\pi/5))$? I have tried setting $x=\zeta_5$ and factoring $$x^5-1=0 \rightarrow (x-1)(x^4+x^3+x^2+x+1)=0 $$ However after ...
2
votes
2answers
251 views

When $mB \neq B$? $m$ is a maximal ideal of $A$, $A \subseteq B$

The following is a question I have asked here without receiving any comments, therefore I post it here: Let $A \subseteq B$ be commutative rings, $m$ a maximal ideal of $A$. When $mB \neq B$? This ...
0
votes
0answers
73 views

semigroups regular!

I am trying to prove that if a maximal group image G(S) of a Clifford semigroup S, is abelian-by-finite does the Clifford semigroup S is abelian-by- finte?
2
votes
1answer
346 views

A particularly “natural” algebraic structure with three commutative, pairwise-distributive operations

EDIT: As mentioned in my answer below, I was mistaken in thinking Dirichlet convolution distributes over ordinary convolution. I'm leaving this question here for reference. I keep stumbling on the ...
0
votes
0answers
14 views

Decompose a multivariate polynomial into a permutation on $F_{2^n}$ and an affine transformation

Let $S(x_1,...,x_n)=(y_1,...,y_n)$ be a secret permutation on $F_{2^n}$. $L$ is a secret $F_{2^n}\rightarrow R^{m}$ affine tranformation. $m$ can be smaller than $n$, while $n$ is ususally less than ...
0
votes
0answers
65 views

Proving the algebraic independence of certain elements

Let $k$ be a field of characteristic zero and $R$ be the polynomial ring $k[x_1,...,x_n,t_1,...,t_n]$. Let $P_i = (a_{i1}:a_{i2}:a_{i3})$ be $n$ points in the projective plane over $k$, such that not ...
-3
votes
1answer
96 views

How to check if argument of complex number is rational multiple of $\pi$ [closed]

Assume there is give a complex number $z\in \mathbb{C}$ of modul 1. Therefore it can written as $e^{i2\pi x}$. What are the known criteria for deciding if $x$ is a rational number???
0
votes
1answer
101 views

Maximal group image!

How to prove that if S is a finitely generated Clifford semigroup its maximal group image is actually the S_{e_{n}}?
4
votes
3answers
301 views

What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
1
vote
2answers
244 views

Subgroup of Projective general linear group on complete discrete valuation ring

Let $R$ be a complete dvr and $k$ its residue field of positive characteristic. Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$. Is there some ...
3
votes
2answers
167 views

Transitivity of discriminant for flat algebras

Sorry if the question doesn't feed this site, I'm reposting it from MSE. Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a ...
0
votes
0answers
73 views

What is the diameter of the Cayley graph of $S_n$ w.r.t the generator $\{(12),(23), . . . ,(n − 1 n) \}$? [duplicate]

Given a symmetric group $S_n $ and the generator set : $S = \{(12),(23), . . . ,(n − 1 n) \}$ is there any closed form expression for the diameter of the Cayley graph generated by this set of ...
1
vote
0answers
149 views

Separability of a simple ring extension

Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
9
votes
3answers
291 views

Real and Quaternionic Representations according to Weights

According to this question, it is easy to know whether a representation is self dual or not: just check if the weight distribution in space is symmetric about the origin. Now, for self dual ...
1
vote
2answers
171 views

What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?

Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...
2
votes
0answers
96 views

Classification of symplectic representations of quaternion division algebras

I would like to know the classification of representations of the form $\rho:B^{\times}\to Sp(V,F)$ or ($Gsp(V)$), where $B$ is a quaternion division algebra over a number field $F$ (or ...
3
votes
0answers
113 views

Does there exist a continuous surjection? [closed]

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
5
votes
1answer
220 views

Can you “combine” Ord and Mon to get Cat?

Mon is the category of moniods, which can be seen as categories with one object. Ord is the category of preorders, which can be seen as categories with up to one morphism in each homset. Is there ...
-1
votes
1answer
173 views

Why do we not lose any generality by proving it only for finitely generated groups [closed]

In the proof of following theorem, in a paper by Farkas- Here $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ and $U_1(\mathbb{Z}G) $ is the set of normalized units of the integral group ring ...
11
votes
2answers
539 views

Obstructions for a group to be the multiplicative group of a field [duplicate]

It is well known that every finite multiplicative subgroup of a field is cyclic. I somehow got interested in a possible reverse implication: Assume we have an abelian group $G$ whose every finite ...
3
votes
2answers
184 views

Solvable Lie algebras: embedded in upper triangular matrices?

Let $K$ be an arbitrary field and $\mathfrak{g}$ a finite-dimensional Lie $K$-algebra. Let $\mathfrak{nil}_n\leq\mathfrak{sol}_n\leq\mathfrak{gl}_n$ be the Lie algebras of all ((strictly) ...
30
votes
2answers
3k views

Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
0
votes
3answers
181 views

Counting zero-sum free sequences of a given length in $\mathbb{Z}_n$

Let $n$ and $d$ be positive integers. Define $\alpha_n^d$ to be the number of vectors $(x_1, x_2, \cdots, x_d)$ in $\mathbb{Z}_n^d$ such that given any subset $S$ of $\{ 1, 2, 3, \cdots d\}$, ...
1
vote
1answer
95 views

The automorphism groups of smallest grammars of a language string are isomorphic

Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = ...
5
votes
1answer
419 views

Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...
3
votes
1answer
124 views

When does a faithful module have an element with zero annihilator?

This is a follow up of Example of a finitely generated faithful torsion module over a commutative ring. Let $M$ be a finitely generated module over a commutative ring $R$ with the property that ...
2
votes
2answers
162 views

What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null

Let $\mathbb F_q$ be the finite field with $q$ elements. Suppose $V$ is a linear space of dimension $n$ over $\mathbb F_q$, and $r<n$. What is the maximal $k$ such that for arbitrary $k$ subspaces ...
1
vote
0answers
58 views

Can we give efficiently the solution of a bilinear system of equations over a finite field?

Consider a finite field $F$ and suppose we have a system of equations $$h_1(\alpha,\beta)=0,h_2(\alpha,\beta)=0,...,h_t(\alpha,\beta)=0$$ where $\alpha=(\alpha_1,...,\alpha_s)$ and ...
1
vote
1answer
73 views

An example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$

In the wake of my curiosity on this kind of things, I was thinking if there is an example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$. Every example I ...
10
votes
1answer
490 views

A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ? This question was first posted here.
1
vote
0answers
39 views

About alternating polynomials an the Rowen's notation

Some definitions... Definition 1: A polynomial $f(X_1,\dots ,X_d)$ is $t$-linear if the variables $X_1,\dots ,X_t,\; t\leq d$ appear in all monomials of $f$ and degree of $X_i,\; i=1,2,\dots ,t$ on ...
5
votes
2answers
330 views

TM and abstract algebra

Usually, during lectures Turing Machines are firstly introduced from an informal point of view (for example, in this way: http://en.wikipedia.org/wiki/Turing_machine#Informal_description) and then ...
3
votes
0answers
198 views

Is the “algebraic closure” of the quaternions, finite dimensional? [closed]

This post is a sequel of: What's the algebraic closure of the quaternions? $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r x^r$, but it is not for the polynomials ...
3
votes
1answer
339 views

A More Advanced Version of Aluffi's Chapter 0

This is a crosspost of this question from MSE. Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce ...
6
votes
1answer
222 views

Action of the homotopy braid groups on reduced free groups

Firstly some definitions: $B_n$ is the braid group with $n$ strands. $\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...
1
vote
1answer
162 views

Does the associated Lie algebra determine a group?

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...
8
votes
2answers
128 views

Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$

This question relates to Realizing groups as automorphism groups of graphs. Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?
29
votes
1answer
1k views

Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$ such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall ...
1
vote
0answers
94 views

Criterion for global dimension of subring

All rings are assumed to be associative and unital. If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...
1
vote
0answers
269 views

Testing the faithfulness of group homomorphisms by testing on the level of induced Lie Algebras

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathrm{gr}_k(G)=\Gamma_G(k)/\Gamma_G(k+1)$ and ...