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

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19
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276 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
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0answers
33 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
votes
0answers
119 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 associativity,"...
5
votes
2answers
188 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 ...
5
votes
0answers
209 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
103 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
1answer
141 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
vote
1answer
137 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
1answer
319 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
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
votes
1answer
250 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
0answers
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
127 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
1answer
222 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
1answer
504 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
0answers
70 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
2answers
143 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
votes
1answer
152 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
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
votes
1answer
247 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
230 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
0answers
78 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
73 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
122 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
1answer
90 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
votes
1answer
388 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
0answers
179 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
147 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
227 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
229 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
votes
0answers
187 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
0answers
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 [...
1
vote
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
294 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
291 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
1answer
109 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
votes
2answers
275 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
0answers
333 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
votes
1answer
93 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, ...
2
votes
1answer
256 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
0answers
104 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 $M_{\Delta}=\...
1
vote
1answer
149 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 ...
8
votes
1answer
260 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
1answer
194 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
0answers
75 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 $\mathcal{A}...
22
votes
2answers
525 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 ...