# Tagged Questions

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votes

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131 views

### The center of a(n endomorphism) ring is a PID

Let $A$ be a torsion-free Abelian group, and let $\mbox{End}(A)$ be its endomorphism ring. Denote by $R\doteq\mathbf{Z}(\mbox{End}(A))$ the center of $\mbox{End}(A)$. What would be necessary and/or ...

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**1**answer

411 views

+50

### A paper by Y. Morita

The corresponding bibliographical details are:
Yoshihito Morita, Elementary proofs of the commutativity of rings satisfying $x^{n}=x$. Mem. Defense Acad. 18 (1978), no. 1, 1–24.
Does anybody here ...

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**0**answers

20 views

### Show that $(X_{k_1}^{a_1},…,X_{k_s}^{a_s})$ is $(X_{k_1},…,X_{k_s})$-primary [migrated]

Show that $I=(X_{k_1}^{a_1},...,X_{k_s}^{a_s})$ is $(X_{k_1},...,X_{k_s})$-primary, where $I$ is the ideal generated by monomials $X_{k_1}^{a_1},...,X_{k_s}^{a_s}$ .$\qquad$ ...

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64 views

### when a prime ideal is maximal differential ideal in a UFD

Is the prime ideal $\langle X^{2}+Y^{2}-1\rangle$ a maximal differential ideal in differential ring $\mathbb{Q}[X,Y]$ with
derivatives $D(X)=Y, D(Y)= -X$?
I know there are maximal ideals like ...

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**0**answers

63 views

### A Lie group associated to a matrix via semi direct product [migrated]

Assume that $A \in M_{n}(\mathbb{R}) $ is a matrix. Then $A$ generates a one parameter (with parameter $t\in \mathbb{R}$) family of group automorphisms of $\mathbb{R}^{n}$ with $x\mapsto ...

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**1**answer

165 views

### Are the positive multiplicative group and the additive group of the field of algebraic numbers isomorphic?

Let $F$ be the field of real algebraic numbers. Is it true that the positive multiplicative group $(F_{pos}^*,\cdot,1)$ is isomorphic to the additive group $(F,+,0)$ (as abstract groups, not ...

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votes

**1**answer

89 views

### An isomorphic invariant in ring theory

Let $R$ be a unital ring. We define the Murray Von Neumann relation $M$ on $R$ as follows:
We say $a M b$ iff $a=xy,\;b=yx$ for some $x,y\in R$. (This is inspired by the usual Murray Von Neumann ...

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**1**answer

110 views

### Existence of class modules for finite groups

I asked the following question on Stackexchange and got no reply so I am reposting it here. Let $G$ be a finite group. A $G$-module C is a class module if, for all subgroups $H \subset G$:
1) ...

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**2**answers

679 views

### Tensor product of fields over integers

Inspired by this question we ask;
Is there a name for each of the following properties about fields? what are some examples other than $\mathbb{Q}$?:
1.A field $K$ with the property that ...

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votes

**1**answer

787 views

### Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?

Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$
We can refer to the elements of $\mathbb{J}$ as "joiners."
The product of joiners is inherited from $\mathbb{Z}$.
The sum of joiners ...

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253 views

### Splitting subspaces and finite fields

Hellow. I'm sure that the following is truth, but I can't prove it.
Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and
$A = ...

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**1**answer

218 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 ...

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**1**answer

130 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
...

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74 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 ...

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274 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 ...

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95 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 ...

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**1**answer

76 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 $\operatorname{Aut}(A\times B) = \operatorname{Aut}(A)\times\operatorname{Aut}(B)$?

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277 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 ...

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**1**answer

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 ...

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**1**answer

97 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 ...

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185 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 = ...

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82 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 ...

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146 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 ...

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156 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 ...

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109 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 $\text{char}\,k > 0$?

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197 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 ...

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148 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 ...

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**1**answer

172 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.
...

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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 ...

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61 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 ...

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**1**answer

114 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 ...

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94 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) ...

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119 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 ...

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386 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 ...

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217 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 ...

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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 $a_{\kappa,\nu}$ ...

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**1**answer

97 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 ...

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546 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 ...

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108 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 ...

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**1**answer

301 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 $, ...

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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 $A \not\cong A \oplus \mathbb{Z}$?

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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, ...

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163 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 $n$ ...

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142 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?

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101 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 ...

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144 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 ...

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**1**answer

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 ...

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**2**answers

286 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 ...

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**1**answer

383 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 ...

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70 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 ...