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63 views

### The normalizer problem for group rings

I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = ...

**2**

votes

**0**answers

39 views

### Assuming $depth M\ge depth N$, what can one say about $depth M_p$ and $depth N_p$?

Definition. Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite R-modules, $p$ a prime ideal, and $I$ an ideal such that $IM\neq M$. Then the common length of the maximal $M$-sequences in ...

**1**

vote

**0**answers

45 views

### Beaumont - Pierce Principal theorem

In book $\text{R. Göbel,P. Hill, Wolfgang "Abelian Group Theory and Related Topics"}$, I found next Beaumont -Pierce Principal theorem: Any torsion-free ring $R$ of finte rank is quasi-equal to ...

**0**

votes

**0**answers

161 views

### Basic question about power series and complete group algebras

This is a pretty basic question, but I suspect it might be too exotic for math.stackexchange.
Let $\mathbb{Z}_p$ be the $p$-adic integers. For free pro-$p$ group $F_r$ of rank $r$, we can consider ...

**3**

votes

**0**answers

72 views

### Freeness of a matrix semigroup

Motivated by some questions in the dimension theory of self-affine sets, a colleague and I are interested in the freeness (or otherwise) of the subsemigroup of $SL_\pm(2,\mathbb{R})$ generated by the ...

**1**

vote

**2**answers

110 views

### Can powers of a maximal ideal stabilize without vanishing?

Let $A$ be a local ring with maximal ideal $m$. Suppose that there exists some positive integer $k$ such that $m^k = m^{k+1}$.
Is necessarily $m^k = 0$ ?
If $m$ is finitely generated, this follows ...

**2**

votes

**0**answers

31 views

### Tensor of Relative Bases

Suppose $U,V,W$ are the cyclotomic fields $\mathbb{Q}(\zeta_{m_3})$, $\mathbb{Q}_(\zeta_{m_2})$, and $\mathbb{Q}(\zeta_{m_1})$ respectively, where $m_1 \mid m_2 \mid m_3$ so that $U/V/W$ is a tower of ...

**3**

votes

**2**answers

100 views

### Covering derivations of a quotient algebra

Let $(\mathcal{A},+,·)$ an algebra and $\mathcal{I}$ an ideal of $\mathcal{A}$.
Is easy to check that if $D\in Der(\mathcal{A})$ with $D(\mathcal{I})\subseteq I$ then $D$ induces a derivation $D_I$ ...

**5**

votes

**1**answer

88 views

### Symmetric algebras of given dimension

Fix an algebraically closed field $F$. Are there only finitely many symmetric algebras with unit over $F$ of a given finite dimension (up to isomorphism)? By symmetric I mean a Frobenius algebra where ...

**3**

votes

**2**answers

118 views

### Rings all of whose torsion modules are cyclic

Let us call a (possibly non-commutative) ring $R$ "very good" if every finitely generated torsion left $R$-module is cyclic. Here is an example of such a ring:
Let $k=\mathbb{C}((t))$ and let ...

**1**

vote

**0**answers

59 views

### Embedding of fields in central simple algebras over number fields

Let $K$ be a non-real CM number field of degree $2d$, with maximal totally real subfield $K_0$, and let $A$ be a central simple algebra over $K$, so that $A\simeq M_n(E)$, the $n\times n$ matrix ring ...

**2**

votes

**1**answer

79 views

### annihilator of minimal prime ideal in a commutative Noetherian ring

Let M be an R-module of finite length and N a maximal submodule of M. Is there an element m in M such that m(N:M)=0? It is an generalization of this result:
In a Notherian ring R, all minimal prime ...

**7**

votes

**0**answers

149 views

### Terminology for vanishing of Hochschild homology with symmetric coefficients?

In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then
it should hopefully be understood by most readers as saying ...

**1**

vote

**0**answers

26 views

### Projectivity of a faithfully balanced self-orthogonal bimodule

Let $_RT_S$ be a faithully balanced self-orthogonal bimodule over a pair of noncommutative rings $(R,S)$, if $_RT$ is projective as a left $R$-module, can we say $T_S$ is also projective as a right ...

**0**

votes

**0**answers

154 views

### Name of some commutative ring akin to $p$-adics

I need help in identifying the naming convention of some commutative ring described below.
Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ...

**2**

votes

**3**answers

316 views

### Subring of ring

Let given ring $R$ without zero divizors, where adittive group of $R$ with zero torsion. Let given subring $R_0\leq R$, and $p$ is prime number, such that $\forall r\in R, \exists i>0 : p^ir\in ...

**-1**

votes

**0**answers

23 views

### localization on the commutative semi prime ring

my question is: Why If $Z$ is a commutative semiprime,then we may assume that n! is invertible in $Z$ by localization at powers n!

**0**

votes

**0**answers

27 views

### on composition of derivation in the commutative domain

Lemma:Let $Z$ be a commutative domain and $d_{1},...,d_{m}$ be such derivations of $Z$ that the identity(equation) $$$$ $ \qquad 0= \sum\limits_{\sigma \in S_{m}} \prod\limits_{i=1}^m ...

**4**

votes

**0**answers

101 views

### Does it follow that any element of $J(A)$ is nilpotent?

Let $A[x]$ be the algebra of polynomials with coefficients in a $k$-algebra $A$. Assume that, for any simple $A[x]$-module $M$, we have $\text{End}_{A[x]} = k$. Does it follow that any element of ...

**2**

votes

**1**answer

78 views

### projective module of rank one over notherian ring

Is finitely generated projective module M of rank one over regular commutative notherian ring free?
Bass (Illinois Math J, 1963) showed that in case M is nonfinitely generated, it is free. I am ...

**-1**

votes

**0**answers

55 views

### How to relate two equations involving composition of the derivations of the associative ring

This question arose while reading the paper On the composition of derivations, by J.Krempa and J.Matczuk.
Let R denote an associative ring For convenience, we will take $chaR=\infty$,f the additive ...

**0**

votes

**0**answers

44 views

### One-sided endomorphism rings of centred bimodules

Let R be an associative unital ring. An R-bimodule M is called centred bimodule if M = R*Z(M), where Z(M)={m:rm=mr,∀r∈R}, i.e., M is generated as an R-module by the set of R-centralizing elements. ...

**2**

votes

**1**answer

128 views

### Finite rank ring

Let given ring $R$ of finite rank. Is it true that for all primes $p$ large enough modules $Der_{\mathbb{Z}}(R/pR) = \{0\}$?
For every ring we define $Der_{\mathbb{Z}}(R)$ as set of linear operators ...

**17**

votes

**0**answers

244 views

### Topological loops vs. algebro-geometric suspension in Hochschild homology

Let $k$ be a base commutative ring, and let $A$ be a (unital but not necessarily commutative) $k$-algebra. The cone on $A$ is the ring $CA$ of infinite matrices $(a_{ij})_{i,j \geq 1}$ that are ...

**1**

vote

**0**answers

30 views

### connected stable rank

There is a beautiful formula by Leonid Vaserstein relating the Bass and topological stable rank of a commutative unital Banach algebra A to
that of the matrix algebra M_n(A). Is there something ...

**1**

vote

**0**answers

61 views

### notation for vector product in the space

The notation for vector (a.k.a. cross) product in $\mathbb{R}^3$ I usually see is $\times$.
However, some places use $\wedge$ instead, which IMHO creates a lot of confusion, as $\wedge$ usually is ...

**2**

votes

**0**answers

112 views

### For f: X -> Y -> X, what is the name of the property whereby for all x in X and y1, y2 in Y, f(f x y1)y2 = f(f x y2)y1?

My PL group has been discussing this (so we are really professional reject mathematicians, but this is more about the math behind what we are doing). Some of us called it a "generalized ...

**5**

votes

**2**answers

168 views

### Lefschetz Principle for semisimplicity

I think I can prove the following using the compactness of first order logic and I am wondering what a purely algebraic proof would look like.
Let $R$ be a unital ring (not necessarily ...

**2**

votes

**1**answer

161 views

### Where does the algebraic closure enter into Block's Theorem?

When applying Block's Theorem on the structure of differentiably simple rings to Lie algebras most authors require an algebraically closed field, but I can see no reference to algebraic closure in ...

**5**

votes

**0**answers

202 views

### A roadmap to learn about finite-dimensional commutative associative real or complex unital algebras

I've always been secretly fascinated with the rich structure and applications of finite-dimensional associative unital algebras over complete fields. In particular, I am very much interested in the ...

**1**

vote

**0**answers

99 views

### A simple Lie algebra with modules of a particular type

I’m trying to copy the following construction of P. Forster in group theory for Lie algebras. He takes a non-abelian simple group E which has an FpE-module V such that R = Rad(V ) is faithful and ...

**3**

votes

**1**answer

95 views

### Is every $n$-ary semigroup a subalgebra of an algebra derived from a binary semigroup?

Let $(A,f)$ be an $n$-ary semigroup ($n \ge 2$). Then there exists a ($2$-ary) semigroup $(\overline A,*)$ with an inclusion homomorphism $A \hookrightarrow \overline A$ such that that the ...

**1**

vote

**1**answer

72 views

### how to write down comatrix of the exceptional Jordan algebra

Suppose we have exceptional Jordan algebra, which is a $3\times3$ matrix ...

**1**

vote

**1**answer

121 views

### A characterization for the ideals of $A+XB[X]$ and $A+XB[[X]]$

Let $A \subseteq B$ be an extension of commutative rings with identity. Then $A+XB[X]$ and $A+XB[[X]]$ are the polynomial and power series rings over $B$ whose constant terms are in $A$. Is there any ...

**3**

votes

**1**answer

235 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

**0**answers

71 views

### What is the relation (if any) between Clifford algebras and Azumaya algebras?

Suppose the base field is $\mathbb{C}$ and the Clifford algebra is the classical one (i.e. associated to a quadratic form in $n$ variables). It seems that there are relations between Clifford algebras ...

**7**

votes

**1**answer

226 views

### Jordan-Hölder-like statements for modules with $\Delta$-filtrations over a quasihereditary algebra

Definitions
Let $A$ be an Artin algebra (for instance, take $A$ to be a finite dimensional algebra over some field) and label the isomorphism classes of simple $A$-modules by the elements of a ...

**3**

votes

**0**answers

387 views

### Some counter examples in group theory

In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in ...

**5**

votes

**2**answers

108 views

### Positive Elements of a $\ast$-Algebra

In a $C^*$-algebra ${\cal A}$, a positive element is a one of the form $aa^*$, for some $a \in {\cal A}$. It is known that the set of positive elements is a cone, and that for $a,b$ two non-zero ...

**7**

votes

**1**answer

199 views

### Fuzzy logic of Godel

In Gödel logic, is conjunction definable from implication, negation , and disjunction?
We know that conjunction in that logic is not definable from negation and implication.

**5**

votes

**1**answer

299 views

### vector spaces with uncountable dimension and a nice basis

Unlike the reals considered as a vector space over the rationals, I know of a number of nice examples of vector spaces with uncountable dimension that have a nice basis.
For example, the space of ...

**2**

votes

**0**answers

59 views

### Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons"
though certainly there are others.
Consider the following ...

**0**

votes

**2**answers

134 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?

**1**

vote

**1**answer

142 views

### Cyclic faithfully flat modules

I am looking for an example of a cyclic faithfully flat $R$-module that is not projective. Could someone help me?

**1**

vote

**0**answers

47 views

### Extremal roots of Bernstein-Sato polynomials

Let $f \in \mathbb{C}[x_1,\ldots,x_n]$.
Consider $D[s]$, where $D$ is the ring of polynomial coefficient differential operators in $n$ variables, and $s$ is an additional formal variable.
Suppose ...

**10**

votes

**1**answer

229 views

### Are all separable algebras Frobenius algebras?

Let $\mathcal C$ be a [added later: semi-simple] tensor category, and let $A=(A,m:A\otimes A\to A,i:1\to A)$ be an algebra object in $\mathcal C$.
The algebra is...
Separable if there is an ...

**5**

votes

**2**answers

213 views

### Dickson/determinant type polynomial (updated)

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation}
...

**2**

votes

**0**answers

69 views

### Morita equivalence of $K$-algebras

Given $K$ a unital commutative ring and $A$ a $K$-algebra different from $K$. Can $K$ be Morita equivalent to $A \amalg A$, where $A \amalg A$ is the coproduct in the category of unital associative ...

**2**

votes

**0**answers

61 views

### Involution of unital based ring (Grothendieck ring of a fusion category)

Let $A$ be a unital based ring in the sense of [Ostrik, arXiv:math/0111139]. As part of the data we have a base $B = \{b_i\}_{i\in I}$, and an involution $i \mapsto \bar i$ of $I$ whose induced map ...

**3**

votes

**1**answer

118 views

### A question about a form of elements of G2

Let $f$ be an automorphism of the octonions algebra.
Then $f(x)=x$ for $x\in \mathbb R$ and $f$ restricted to $Im \mathbb O$ is in $SO(7)$.
By the properties of the rotations there is an orthonormal ...