The tag has no usage guidance.

learn more… | top users | synonyms

0
votes
1answer
186 views

Order-Perserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
72
votes
2answers
3k views

$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
-1
votes
0answers
31 views

the term “minimal degree” in a set [closed]

I have been reading a proof to the division algorithm of polynomials in a ring. I am struggling to understand the portion of the proof in regards to how the remainder polynomial if it is not zero, it ...
4
votes
1answer
113 views

An example of an Azumaya algebra that isn't free over its centre

Azumaya originally defined an Azumaya algebra (which he called a proper maximally central algebra) to be an algebra A which is a free module of finite rank over its centre Z such that the natural map ...
0
votes
0answers
69 views

the linear span of all matrix coefficients is $C(G,\mathbb{C})$ where $G$ is a finite group

Theorem. Let $\{(R_{\alpha},V_{\alpha})\}$ be a complete set of inequivalent irreducible finite dimensional representations of a finite group $G$. Let $V_{R_{\alpha}}$ be the subspace generated by all ...
7
votes
1answer
267 views

For a new operation on a finite group of odd order giving a loop structure, when does this also gives a group

For finite groups $G$ of odd order, as $x \mapsto x^2$ is bijection (but no automorphism in general) then, we can define for each $g \in G$ the element $x^{1/2}$ by requiring $(x^{1/2})^2 = x$. Then ...
53
votes
1answer
1k views

$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$? This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
3
votes
2answers
89 views

About Kurosch-Ore theorem

Where can I find the proof of Kurosch-Ore theorem in lattice theory? The statement of this theorem is: Let $L$ be a modular lattice with $0$ and $1$ that satisfies both chain conditions. Then for any ...
-2
votes
1answer
55 views

On group of automorphisms of direct product of nonabelian finite groups [closed]

Let $A, B$ be finite nonabelian groups such that $(|A|, |B|) = 1$. Is $\operatorname{Aut}(A\times B) = \operatorname{Aut}(A)\times\operatorname{Aut}(B)$?
3
votes
0answers
271 views

A relation between a ring with its polynomial ring

Let $\{f_i(x)\}_{i\in I}$ be a subset of $R[x]$ where $R[x]$ is the polynomial ring of $R$(a commutative ring with identity). If the ring $R/\langle f_i^2(n)-f_i(n)\rangle_{i\in I, n\in A}$, for every ...
-1
votes
1answer
86 views

Notation for a Homomorphism [closed]

Is there a (common) notation which denotes a function, $f$, to be a homomorphism? I have found myself writing, "let $f: X \rightarrow Y$ be a homomorphism" several times. This is fine, but I would ...
5
votes
1answer
180 views

Description of the algebra of $G$-invariant polynomials by generators and relations

Fix $n > 1$ and let $\zeta \in \mathbb{C}$ be a primitive $n$-th root of unity. Let $G \subset \text{SL}_2(\mathbb{C})$ be a cyclic subgroup of order $n$ generated by the diagonal matrix $g = ...
-1
votes
1answer
79 views

Group associated to the monoid $({\cal P}(X\times X), \circ)$

Consider the set ${\cal P}(X\times X)$. It can be endowed with a binary operation $\circ$ where $$A\circ B = \{(a,b)\in X\times X:\exists z\in X((a,z)\in A\land (z,b)\in B)\}.$$ Note that ...
4
votes
0answers
76 views

$SL_2(\mathbb{F})$, decomposing $\mathbb{C}\{X\}$ into irreducible $G$-representations and dimensions [closed]

Let $\mathbb{F}$ be a finite field with $q$ elements and $H = \mathbb{F}^\times$, the multiplicative group of $\mathbb{F}$. It is known that $H$ is a cyclic group of order $q - 1$, so $\widehat{H} = ...
8
votes
0answers
150 views

$A$-module is free if and only if equation involving Hilbert-Poincaré series holds, $M$ infinitely generated case

See my question here. Let $A = \oplus_{i \ge 0} A_i$ be a nonnegatively graded commutative algebra and $M$ a nonnegatively graded $A$-module. Assume in addition that $A_0 = k$ and all vector ...
8
votes
0answers
139 views

$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...
7
votes
1answer
88 views

Center of $k$-algebra with two generators and sole defining relation $yx - xy = 1$ when $\text{char}\,k > 0$

Let $A(k)$ be a $k$-algebra with two generators, $x$, $y$, and one defining relation: $yx - xy = 1$. What is the center of the algebra $A(k)$ in the case $\text{char}\,k > 0$?
1
vote
2answers
194 views

Is $K\cap \langle H\cup N\rangle‎\subseteq‎ \langle H\cup (K\cap N)\rangle$? [closed]

If $G$ is a arbitrary group, $H,K,N\leq G$ such that $H‎\subseteq‎ K$, then $K\cap \langle H\cup N\rangle‎\subseteq‎ \langle H\cup (K\cap N)\rangle$?. If it is not true, how can I find an ...
4
votes
1answer
146 views

Obstruction for two subgroups to be conjugated by an automorphism

Altough this sounds as a very basic question, I didn't receive any answer on stack exchange and by people more knowledgeable than me Take $p$ a prime number and $P$ an abelian finite $p$-group. Let ...
0
votes
1answer
159 views

How to solve this system of equations? [closed]

I am trying to find a Poisson bracket on an algebra, and need to find a solution to a system of equations. The system of equations is very complicated, with more than 10000 equations and 60 variables. ...
2
votes
1answer
81 views

How can the Cayley table for the elements of basis of a Cayley-Dickson algebra be summarized in an algebraic expression?

One would be able to construct a Cayley table that has all $e_i$ elements of the basis of algebra $A$ where $0<i<\dim A$ such that $e_0=1$, $e_1=i$, $e_2=j$ and so on. I'm looking for an ...
2
votes
0answers
148 views

Problem in abstract algebra [closed]

Let $p$ and $q$ be two distinct primes. For a field $F$, assume that $\deg(\alpha, F)=p$. Is it necessarily true that $\deg(\alpha^q, F)=p$? Is there any counterexample? It is not an exercise problem ...
0
votes
0answers
58 views

“4th order” floretions- cyclic transformation question

In response to the last paragraph mentioning "swapping" operations in this post, I would like to mention what the reference is to and one question I currently have. Assume $X = abCD$ is some 4th ...
4
votes
4answers
459 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 ...
2
votes
3answers
106 views

Non-principal prime ideals in infinite distributive lattices

Given an infinite distributive lattice $L$, does $L$ contain a non-principal prime ideal $I$, or a non-principal prime filter $F$? ($I$ is said to be principal if there is $x\in L$ such that $I=\{y\in ...
3
votes
1answer
109 views

Bibliography suggestion for Kummer theory

I already posted a question about a sum involving the degree of a Kummer extension. Now I'm interested in a more specific fact about Kummer extensions. From Hooley's paper "On Artin's conjecture", we ...
4
votes
1answer
93 views

Maximal cyclic quotient of a $p$-group

Let $G$ be a finite abelian $p$-group, $p$ a prime. I say that a pair $(G',\varphi)$ is a maximal cyclic quotient (please excuse me if this definition already exists and refers to a different concept) ...
2
votes
1answer
383 views

When does $R [x]/I $ has infinitely many idempotents?

Let $R$ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotents and an unextended ideal $I$ in $R[x]$ such that $R[x]/I$ has ...
5
votes
2answers
544 views

Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $Gal(T(f))$ $\simeq$ The commutator subgroup of $Gal(f)$?

I asked this question at MSE but I did not receive an answer. So I ask it at MO: We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is denoted by ...
11
votes
1answer
530 views

Elementary proof for Hilbert's irreducibility theorem

I have tried to find a complete proof for Hilbert's irreducibility theorem, but everything I found was way beyond my level of understanding. I am only interested in the simple case where the ...
5
votes
0answers
216 views

Endofunctors on the category of groups which are Galois- related to a linear map on $\mathbb{Q}[x]$

I thank J.C. and Arturo Magidin for interesting suggestions in the comments to this question In this post the field of rational numbers is denoted by $\mathbb{Q}$. The space of polynomials with ...
0
votes
1answer
90 views

Natural Poisson brackets on $S(V^*)$

Let $V$ be a finite dimensional vector space and $S(V)$ the corresponding symmetric algebra. Suppose that we have a Poisson bracket $\lambda = \{,\}: S(V) \otimes S(V) \to S(V)$. Let $V^*$ be the dual ...
2
votes
0answers
97 views

Is there any construction of infinite dimensional algebraic division ring? [closed]

I know that there is a division algebra over $\mathbb{Q}$ such that it is algebraic and infinite dimensional over it's center i.e. $\mathbb{Q}$. But for construct this division algebra. we can use ...
2
votes
0answers
134 views

calculation in a group ring

I have some problems with the verification of the third equation in Lemma 1 in this paper. First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ ...
3
votes
1answer
262 views

Maximal ideal of group ring

Let $R$ be a finite commutative ring with identity and $G$ an finite abelian group. Is there any more conditions (on $R$ or on $G $) under which we can characterize maximal ideals of group ring $RG $, ...
2
votes
0answers
76 views

Polynomial constraints triggered by irreducibility [closed]

I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic: $$af^2 + bf + c = 0$$ If we're working in a ring, ...
0
votes
2answers
139 views

Making idempotent element by a relation [closed]

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?
4
votes
0answers
161 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 $n$ ...
3
votes
0answers
98 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
139 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 ...
0
votes
1answer
135 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}}?
0
votes
1answer
129 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
2answers
276 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 ...
28
votes
6answers
4k views

Do convolution and multiplication satisfy any nontrivial algebraic identities?

For (suitable) real- or complex-valued functions f and g on a (suitable) abelian group G, we have two bilinear operations: multiplication - (f.g)(x) = f(x)g(x), and convolution - (f*g)(x) = ...
2
votes
1answer
375 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
69 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 ...
12
votes
1answer
276 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 ...
1
vote
2answers
267 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 ...
4
votes
2answers
184 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
74 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 ...