**3**

votes

**2**answers

129 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 ...

**1**

vote

**0**answers

58 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 ...

**-1**

votes

**0**answers

54 views

### Modules over polynomial algebras [on hold]

Let $M$ be a cyclic torsion module over the polynomial ring $k[x_1, \dots, x_n]$.
Let $M = K/J$, where $K$ denotes the polynomial ring $k[x_1, \dots, x_n]$.
Is it true that $J \cap k[x_i] \ne 0$ for ...

**4**

votes

**0**answers

195 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

88 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

85 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

70 views

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

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

**0**

votes

**1**answer

101 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

220 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

62 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

211 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

382 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 ...

**4**

votes

**2**answers

93 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 ...

**5**

votes

**1**answer

182 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.

**2**

votes

**0**answers

56 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

132 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

138 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

43 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

220 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

204 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}
...

**1**

vote

**0**answers

67 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

54 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 ...

**1**

vote

**1**answer

86 views

### Independent set of relations in an algebra [closed]

Let $k⟨X⟩$ be a free associative algebra generated by a set $X$ over a field $k$. Let $S$ be a set of $k$-algebra relations. Then what does it mean by the set of relations are independent ?

**2**

votes

**1**answer

363 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 ...

**4**

votes

**0**answers

156 views

### Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...

**2**

votes

**0**answers

48 views

### question about the group completion of a simplicial monoid

In Daniel Quillen, On the group completion of a simplicial monoid pdf, MIT preprint 1971, Memoirs of the AMS vol 529, 1994, pp. 89-105, I do not understand the following part with question mark
...

**6**

votes

**1**answer

136 views

### presentations of subalgebras

Assume that I have a finitely presented algebra $A$ over the complex numbers (by which I mean that $A$ is generated over $\mathbb{C}$ by finitely many elements $x_1,...,x_n$ subject to finitely many ...

**0**

votes

**1**answer

213 views

### Determinants of tensors [closed]

Consider a tensor of dimension $[d]\times[d]\times[d]$ which is symmetric with respect to every permutation of the indices. Are there any $\textbf{explicit}$ formulas for notions like determinant-like ...

**7**

votes

**1**answer

188 views

### In modules over finite rings $Rv = Rw \iff R^\times v = R^\times w$

Let $R$ be a finite ring (with unit, possibly non-commutative), and $M$ a left module over $R$. Let $v,w\in M$. Then
$$Rv = Rw \iff R^\times v = R^\times w.$$
This follows from Lemma 6.4 in ...

**3**

votes

**0**answers

174 views

### Ring epimorphisms, and epimorphism in the category of small preadditive cats

This question is related to this other question I have asked some time ago. Let $R$ and $S$ be two rings and let $\phi:R\to S$ be a ring homomorphism.
It is well-known that $\phi$ is an epimorphism ...

**0**

votes

**0**answers

53 views

### Writing a module as a direct sum

Let $q_1, q_2, q_3 \in \mathbb{Z}[x,y]$ such that $q_1, q_2$ are algebraically independent and let $S$ be an algebra generated by $q_1, q_2, q_3$ over $F_p$. If writing $S$ as a module over $F_p ...

**1**

vote

**0**answers

119 views

### Intersections of ideals and nilpotence

Let $R$ be a polynomial ring over a field $k$, $R = k[x_1, \dots, x_n]$. Suppose $R'$ is an associative $R$-algebra and it has the property that there exists a degree $m<n$ monomial in the $x_i$'s ...

**5**

votes

**1**answer

257 views

### Does the following $ C^{*} $-algebraic result have a purely algebraic proof?

While studying the proof of Bott periodicity for operator $ K $-theory in this set of notes, I learned this fact:
Theorem. Let $ A $ and $ B $ be $ C^{*} $-algebras. Let $ f,g: A \to B $ be $ * ...

**10**

votes

**1**answer

271 views

### Non-field example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$

I asked this question on math.stackexchange a month ago with no progress, even after a bounty. I hope to eliminate one if the other receives a satisfactory answer.
For an ideal $I\lhd R$ in a ...

**2**

votes

**1**answer

107 views

### Do representations of the universal enveloping algebra $\mathrm{U}\mathfrak{su}_2$ retain the Hopf algebra structure?

A Lie algebra $\mathfrak{g}$ generates its universal enveloping algebra $\mathrm{U}\mathfrak{g}$, which has the structure of a Hopf algebra. Modules of $\mathrm{U}\mathfrak{g}$ are exactly the of ...

**0**

votes

**0**answers

84 views

### The universal property of ring of big Witt vectors

Let $X_1,X_2,\cdots$ be infinite many indeterminates. Define
\begin{equation*}
W_n = \sum_{d\mid n} d X_d^{n/d}.
\end{equation*}
A big Witt vector over a commutative ring $R$ is a sequence ...

**2**

votes

**2**answers

188 views

### Generalizing the commutator and anti-commutator

I was wondering if there's any attempt to generalize the commutator for something general for more than two terms.
Here's what I was thinking of, for $[A,B]=AB-BA$, so for three terms:
$[A,B,C] = ...

**7**

votes

**0**answers

275 views

### Can every commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

All rings here are associative, commutative and unital. By a ring of characteristic zero (resp. of characteristic $p$, for prime $p$) I mean a ring $A$ such that the canonical homomorphism $\mathbb ...

**3**

votes

**1**answer

90 views

### Can you test flatness on $FP_3$-modules?

Let $A$ be a ring, and let $M$ be a right $A$-module. Then $M$ is flat if and only if for each left $A$-module $N$ we have that $Tor^1_A(M,N) = 0$. Becasuse $Tor$ commutes with filtered direct limits, ...

**1**

vote

**1**answer

176 views

### Counting Roots of Unit

Let $p\left( x\right) =%
%TCIMACRO{\tprod \limits_{k=1}^{m}}%
%BeginExpansion
{\textstyle\prod\limits_{k=1}^{m}}
%EndExpansion
\left( x^{e_{k}}-\omega_{k}^{e_{k}}\right) $ be a polynomial with
...

**1**

vote

**0**answers

97 views

### cohomlogy of Diagonal ring

Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let ...

**1**

vote

**1**answer

97 views

### Direct limit and finite presentation of modules

Let $R$ be a ring. Recall that a module $M$ is called finitely presented if there is an exact sequence
$R^n \to R^m \to M \to 0$.
with $n,m \in \mathbb{N}$.
A well known result states that any ...

**7**

votes

**1**answer

251 views

### For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$?

Let $k$ be a commutative ring. Feel free to assume it's a field.
Let $X$ be a set. This question is only interesting when $X$ is infinite.
Write $k^X$ for the $k$-algebra of functions $X \to k$, ...

**1**

vote

**1**answer

187 views

### Is a prime index inclusion of finite groups, separating?

Let $(H \subset G)$ be an inclusion of finite groups.
Let $\{ g_i \ \vert \ i \in I=[1, \dots ,n] \}$ a subset of $G$ of double coset representatives, i.e. $$G = \coprod_{i \in I} Hg_iH$$
On the ...

**2**

votes

**0**answers

73 views

### When the Lie algebra of matrices with zero last rows is Frobenius?

Let $\mathcal{A}_{n,k}$ be the Lie algebra of $n \times n$ matrices over $\mathbb{C}$ for which the last $k$ rows are equal to zero. Suppose that $k$ does not divide $n$. How to prove that ...

**19**

votes

**2**answers

421 views

### What are applications of commutativity theorems for rings?

Herstein's little book "Noncommutative Rings" has a chapter called Commutativity Theorems in which he proves results like Jacobson's theorem: if a ring (associative with identity, please) has the ...

**15**

votes

**1**answer

439 views

### How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, … algebras?

There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique.
What about just traces on separate algebras? That is, take one of ...

**2**

votes

**0**answers

43 views

### Locally free sheaves of algebras vs. algebra bundles

It is well known that there is a bijective correspondence between locally free sheaves of modules and vector bundles (cf. ...

**4**

votes

**2**answers

292 views

### Relation between Associative algebra and group algebra

Let $A$ be an associative algebra over a filed $k$.
Q) What are the condition we can impose on $A$ such that there exists a $G$ such that $A=k[G]$, the group algebra generated by $G$?
I am ...