Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups.

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179 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
3answers
325 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 $\forall r\in R, \exists i>0 : p^ir\in R_0$...
9
votes
1answer
234 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 $\text{End}_{A[x]} M = k$. Does it follow that any element of $J(...
2
votes
1answer
94 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 ...
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0answers
50 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. ...
0
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0answers
109 views

Quotient modules of polynomial rings by maximal one-sided ideal

Let $R[X]$ be a ring of polynomials over an associative unital ring $R$ which is not necessarily commutative. Let $M$ be a maximal left ideal in $R[X]$. It is easy to see that if the intersection of $...
2
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1answer
205 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 $...
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278 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 ...
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0answers
34 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
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0answers
64 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
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0answers
151 views

For $f: X \to (Y \to X)$, what is the name of the property whereby for all $x\in X$ and $y_1, y_2 \in Y,$ $f(f(x,y_1), y_2) = f(f(x, y_2),y_1)$?

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 associativity,...
5
votes
2answers
189 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
1answer
171 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 ...
2
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0answers
114 views

Torsion ideal in symmetric algebra

Let $D$ be a commutative domain, $M$ a $D$-module without torsion, and $S(M)$ its symmetric algebra. Is the $D$-torsion ideal of $S(M)$ the prime ideal of $S(M)$?
6
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211 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
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0answers
104 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
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1answer
144 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 restriction ...
1
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1answer
77 views

how to write down comatrix of the exceptional Jordan algebra

Suppose we have exceptional Jordan algebra, which is a $3\times3$ matrix $X=\left(\begin{matrix}x_1&\phi_1&\phi_2\\\bar{\phi_1}&x_2&\phi_3\\\bar{\phi_2}&\bar{\phi_3}&x_3\\\end{...
1
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1answer
138 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
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1answer
334 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
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97 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
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1answer
253 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
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404 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 \{1,2,\ldots,...
5
votes
2answers
129 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
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1answer
224 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
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1answer
541 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
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0answers
73 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 ...
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2answers
144 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?
2
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1answer
154 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
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0answers
55 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 $P(...
11
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1answer
250 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 $A$-$A$-...
5
votes
2answers
232 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
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0answers
79 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 $K$...
2
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0answers
77 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
1answer
123 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
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1answer
91 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 ?
3
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1answer
390 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
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0answers
188 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 ...
6
votes
1answer
148 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
1answer
228 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 ...
6
votes
1answer
232 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 Hyman ...
3
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0answers
188 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 ...
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58 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 [...
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0answers
143 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 ...
8
votes
1answer
295 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
1answer
292 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
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1answer
111 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 ...
3
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2answers
277 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
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0answers
338 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 Z\...
3
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1answer
95 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, ...