Questions tagged [ra.rings-and-algebras]

Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.

Filter by
Sorted by
Tagged with
4 votes
1 answer
226 views

Finite type injective ring map between domains preserves the open point $(0)$

I am looking for a proof of the following statement without using full power of Chevalley's theorem on constructible sets. We say a domain $A$ is $0$-open if $\{(0)\}$ is open in $\operatorname{Spec}(...
William Sun's user avatar
0 votes
2 answers
129 views

Examples of isomorphic non-equivalent twisted group algebras

Let $F$ be a field, $G$ be a finite group and $\alpha \in Z^2(G, F^*)$ . The twisted group algebra $F^{\alpha}G$ is a $F$-algebra with $F$ vector basis, $\{\bar g : g \in G \},$ and multiplication ...
Eloon_Mask_P's user avatar
2 votes
0 answers
102 views

Filtrations and Koszul algebras

I was looking at this question and asked my self the following: Let $A$ be graded algebra, which is also an $\mathbb{N}_0$-filtered algebra. If its associated graded algebra $\mathrm{gr}(A)$ is ...
Didier de Montblazon's user avatar
1 vote
1 answer
122 views

Polynomial identities satisfied by the Weyl algebra in prime characteristic

The rank $n$ Weyl $A_n(\mathsf{k})$ algebra over a field $\mathsf{k}$ of zero characteristic does not satisfies any polinomial identity. If it were a PI-algebra, Kaplansky theorem would apply (since ...
jg1896's user avatar
  • 2,683
9 votes
2 answers
598 views

Smallest faithful matrix representation of the exterior algebra

Let $R = \Lambda \mathbb{C}^n$ be the exterior algebra on $\mathbb{C}^n$ for some positive integer $n$. It is an associative (graded-commutative) algebra of $\mathbb{C}$-dimension $2^n$. Suppose we ...
benblumsmith's user avatar
  • 2,831
1 vote
0 answers
58 views

Structure of tame concealed algebra of Euclidean type

I wanted to know some references where people have studied the representation theory of tame concealed algebra of Euclidean type. What do we know about the structure of their module category? What ...
It'sMe's user avatar
  • 767
1 vote
0 answers
114 views

Classifying of low-dimensional Frobenius algebras

Does there exist a classification of finite-dimensional Frobenius algebras in low dimensional cases. This question is motivated by the following analogous question for Hopf algebras.
Didier de Montblazon's user avatar
4 votes
1 answer
306 views

Determining when quotient of a polynomial ring is a Gorenstein ring

I would like to be able to look at the ring $R=\mathbb{Z}[x_1,x_2,\ldots,x_n]/\mathcal{I},$ where $\mathcal{I}$ is generated by a finite number of monomials and say whether $R$ is a Gorenstein ring. ...
Haldot's user avatar
  • 215
2 votes
1 answer
150 views

When are elements of a (perfect) semidirect product simple commutators?

I am migrating this question from math stackexchange... I have a semidirect product and I have shown that it is perfect. However, I would like to know whether every element is a simple commutator (...
Makenzie's user avatar
1 vote
0 answers
103 views

Germs of holomorphic functions and invariant functions

Consider a complex vector space $V \cong {\mathbb C}^n$. Consider the ring of germs of holomorphic functions ${\mathcal O}_0 (V)$ at $0\in V$. We know that this ring is Noetherian. Now consider a ...
UVIR's user avatar
  • 971
2 votes
0 answers
127 views

Algebra of finite width matrices

$\DeclareMathOperator\FWM{FWM}\DeclareMathOperator\End{End}$For any ring $R$ there's an algebra of finite width matrices with entries in $R$. By finite width matrices I mean the ones that have only ...
Denis T's user avatar
  • 4,371
7 votes
1 answer
261 views

Non(skew)commutative Lie algebras?

The Lie operad $\text{Lie}$ is generated by a binary operator $[\ ,\ ]$, modulo a degree two relation (skew commutativity $[x,y]=-[y,x]$) and a degree three relation (Jacobi $[x,[y,z]]+[y,[z,x]]+[z,[x,...
Pulcinella's user avatar
  • 5,507
6 votes
1 answer
241 views

Quantum exterior algebra

In Generalisation of the quantum exterior algebra the quantum exterior algebra is discussed: $$ K\langle x_1,\dotsc x_n\rangle/(x_i^2,x_i x_j + q_{i,j}x_j x_i), $$ with nonzero field elements $q_{i,j}...
László Szabados's user avatar
3 votes
1 answer
138 views

Profinite completion of Baumslag-Solitar group as a profinite HNN-extension

I apologize if this question is basic in some sense. I was looking for an example of a non-proper HNN-extension and I found this. In the comments, markvs mentioned the Baumslag-Solitar group $B(2,3)$. ...
Lucas's user avatar
  • 289
0 votes
0 answers
88 views

What is $(C, D, \delta, \gamma)$ and $(C, \delta; D, \gamma)$ Desarguesian?

A projective plane is $(C, \gamma)$-Desarguesian if for any 2 triangles $A_1 B_1 C_1, A_2 B_2 C_2$ in perspective from $C$ (which means $C \in A_1 A_2, B_1 B_2, C_1 C_2$) such that $A_1 B_1 \cap A_2 ...
Display name's user avatar
1 vote
0 answers
93 views

Does the center of any finitely generated associative algebra over a field have finite type?

Consider the monoid algebra $R:=K\langle x_1,\dots,x_n\rangle$ generated by $n$ letters $x_1,\dots,x_n$ for $n>1$ over field $K$. Equivalently, $R$ is the tensor algebra $T(V)$ on the $n$-...
GiS's user avatar
  • 321
4 votes
1 answer
133 views

Kernels of actions on truncated polynomial algebra

Let $p$ be an odd prime, and let $k=\mathbb{F}_p$ be the field with $p$ elements. Let $G=\text{GL}_n(k)$. The group $G$ acts on the truncated polynomial algebra $A:=k[x_1,\ldots, x_n]/(x_1^p,\ldots, ...
Ehud Meir's user avatar
  • 4,969
-3 votes
1 answer
176 views

can the square of a polynomial be written as a sum of squares of polynomials with only even degree terms?

Let $f$ be a polynomial with real coefficients in several indeterminates $x_1, \dots, x_n$. Suppose that $$ f = g^2 $$ for some polynomial $g$. Is it true that we can find polynomials $h_1, \dots, h_m$...
Colin Tan's user avatar
  • 251
3 votes
1 answer
162 views

$\Omega$ for noetherian semiperfect rings

Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$. Let $\Omega^n(mod A)$ be the category of $n$-...
Mare's user avatar
  • 25.7k
-7 votes
1 answer
228 views

Is the Klein group related to the Klein bottle? [closed]

Is the group of symmetries of the rectangle-not-square related to the Klein bottle mathematically? The reason I am asking is because I want to put a Klein bottle coffee cup in a joke about V_4 and ...
Erin Carmody's user avatar
3 votes
0 answers
116 views

Finite algebras with finitely many automorphisms

Let $B'/B$ be a finite locally free algebra. Locally in $B$, there is an isomorphism of $B$-modules $B'\simeq B^{\oplus n}$. When is the automorphism group of $B'/B$ finite? When is it unramified? Is ...
Leo Herr's user avatar
  • 1,084
0 votes
0 answers
156 views

Numerical approaches to functional equations

I'm interested in finding numerical approaches to solving functional equations such as f(xy)=f(x)+f(y), where the equations had no derivatives or integrals, and contains arguments involving x and y . ...
Doug Brunson's user avatar
1 vote
0 answers
119 views

$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?

We have the following identities: $\sin(\frac{\pi}{1})=0$ $\sin(\frac{\pi}{2})=1$ $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ $\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$ Lets start with a definition. Rules ...
mick's user avatar
  • 721
1 vote
0 answers
77 views

tensor dimension/reshaping group

Consider an $N$ dimensional tensor $T$ using the strided view representation used by PyTorch, i.e. we have a storage vector $S$ projected into $N$ dimensions using a size tuple $s$ and a stride tuple $...
mikeyd's user avatar
  • 11
3 votes
0 answers
111 views

A theory of refined h- and f-polynomials for the permutahedra, associahedra, noncrossing partitions, and tropical Grassmannians (references)

Looking for references (insights) on a theory encompassing a notion of refined face polynomials and their associated refined h-polynomials that are generalizations of the relation between ordinary f-...
Tom Copeland's user avatar
  • 9,897
3 votes
1 answer
130 views

Does Noetherianity imply division theorem?

I am trying to understand something which is probably basic for experts so I am sorry if this is not suited for this forum. Let $\mathcal{O}_n$ denote the ring of germs at $0 \in \mathbb{R}^n$ of real-...
cs89's user avatar
  • 971
2 votes
1 answer
181 views

n-ary (polyadic) group "defined for tuples"

Consider an n-ary (polyadic) group, which is a generalization of the "usual" ("binary") groups to n-tuples. The definition requires appropriate standard generalizations of ...
P. Trinli's user avatar
0 votes
0 answers
158 views

When does this commutative non-associative algebra have nilpotent elements?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients $(a_0, \dotsc, ...
mick's user avatar
  • 721
3 votes
0 answers
88 views

Cohn's localization for rings with enough idempotents

I am in the following situation: I have a non-unitary (associative) ring $R$ with enough idempotents or, if you prefer, a small pre-additive category. Actually, I even know that $R$ is right coherent (...
Simone Virili's user avatar
4 votes
1 answer
226 views

Classification of simple modules for the free algebra

Let $A=K\langle x,y\rangle$ be the free associative algebra in two generators over a field $K$ (we can assume that the field is algebraically closed or even $K=\mathbb{C}$ first if that helps) ...
Mare's user avatar
  • 25.7k
7 votes
1 answer
212 views

Density of extended Mersenne numbers?

Consider the subset of odd positive integers defined and constructed as follows by these rules : A) $1$ is in the set. B) if $x$ is in the set , then $2x + 1$ is in the set. C) if $x$ and $y$ are in ...
mick's user avatar
  • 721
3 votes
1 answer
113 views

Equivalent definitions of pro-unipotent coalgebras

I'm trying to find a reference in the literature for equivalence of the following two definitions of pro-unipotent coalgebras. Definition Let be $H$ a coagumented coalgebra and let $\Delta \colon H \...
Liz Nesterova's user avatar
1 vote
0 answers
101 views

Inclusion between rings after localization

Let $\phi:A \to B $ an injective finite ring map between two noetherian integral domains $A,B$. Let $ C \subset B$ a subring of $B$ and assume that there exist a prime ideal $\mathfrak{p} \subset A$ , ...
user267839's user avatar
  • 5,938
0 votes
0 answers
36 views

Almost division ring but only one-side distributive

I am looking for an example of an algebraic structure that is almost a division ring, but is only left-distributive and not right-distributive. Gérard Lang
Gérard Lang's user avatar
  • 2,617
29 votes
2 answers
3k views

Addition and multiplication are commutative but exponentiation and tetration are not. Do we know why?

I was thinking about the idea that succession, addition, multiplication, exponentiation, tetration and so on form a sequence of operations where each is defined as a repeated self application of the ...
Rorsa's user avatar
  • 873
5 votes
2 answers
2k views

Lang's "Algebra" as a self-study book

I am an undergrad senior math major taking a gap year looking to become an actuary. However, I still want to continue learning pure math. I've been looking for a relatively high level text to self ...
11 votes
9 answers
1k views

What are examples of problems we know how to solve for primes (or prime powers), but not for composites?

I am interested in seeing examples of research problems which fall into one of the two following categories: A problem which is solved in the case of primes (or prime powers), but which remains open ...
4 votes
1 answer
100 views

Property of simplicity and semi-simplicity under base change of base field

Suppose $K$ is a field of characteristic $0$ and $A$ is a $K$-algebra. Let $F$ be a field extension of $K$ and let $M$ be an $A$-module. What can we say about simplicity or semi-simplicity of $A_F$-...
amir hossein Ekhlasi's user avatar
4 votes
3 answers
387 views

Can nonnegative functions $f(x,y,z)$ be written as a product of pairwise functions $u(x,y) v(y,z) w(x, z)$?

In my course on probabilistic graphical models, my professor made a claim which I find a little sus. In discussing the equivalence between Markov Random Fields and Factor Graphs, the following example ...
bucket's user avatar
  • 85
1 vote
1 answer
143 views

Invariant ring of the subvariety

Let $G$ be a linearly reductive algebraic group and $X$ be an affine $G$-variety over an algebraically closed field $\mathbb{K}$. Let $Y\subset X$ be a (closed) affine subvariety of $X$ which is also $...
It'sMe's user avatar
  • 767
5 votes
0 answers
217 views

Which properties of the pullback/restriction functor imply surjectivity of a ring homomorphism

Let $f:R\to S$ be a homomorhpism of (noncommutative) algebras over some field k (say the complex numbers) and let $F:Rep(S)\to Rep(R)$ be the corresponding functor between the categories of finite-...
Simon Lentner's user avatar
2 votes
1 answer
101 views

Proof of restrictableness of Lie algebra without basis

$\DeclareMathOperator\ad{ad}$According to the Strade and Farnsteiners' textbook, a Lie algebra $L$ is restrictable if and only if $\ad(L)$ is $p$-subalgebra of $\operatorname{Der}(L)$. Also, the ...
Frank Voigt's user avatar
69 votes
28 answers
7k views

Results from abstract algebra which look wrong (but are true)

There are many statements in abstract algebra, often asked by beginners, which are just too good to be true. For example, if $N$ is a normal subgroup of a group $G$, is $G/N$ isomorphic to a subgroup ...
4 votes
0 answers
156 views

Definitions of torch ring

Well, I find myself in a tangled web of references, definitions and doubt. How do I get myself into these things? Get ready for a rollercoaster of definitions. An FGC ring is a commutative ring whose ...
rschwieb's user avatar
  • 1,593
2 votes
1 answer
69 views

How to define inverse of a non-degenerate triangular structure?

Let $\mathfrak {g}$ be a non-degenerate triangular Lie bialgebra with the non-degenerate triangular structure $r \in \bigwedge^2 \mathfrak {g}.$ Then how does it induce $r^{-1} \in \bigwedge^2 \...
Anil Bagchi.'s user avatar
4 votes
1 answer
211 views

Classifying indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$

I'm now interested in classifying the indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$ : the group ring of $\mathbb{Z}/p\mathbb{Z}$ over the ring $\mathbb{Z}/p^{2}\...
gualterio's user avatar
  • 1,043
0 votes
2 answers
335 views

Torsion of modules

Given a left module $M$ over a domain $R$, I am interested in irreducible elements $r\in R$ such that $r\cdot m=0$ for some $m\in M-\{0\}.$ I think "torsors" would be perfect name for such $...
Adam's user avatar
  • 2,370
3 votes
1 answer
376 views

What is the name for algebras generated by elements, all of whose cubes vanish?

Given a ring $R$ with identity $1$, we can define the exterior algebra of order $k$ over $R$ to be the algebra over $R$, generated by elements $x_1, \dots, x_k$ satisfying $x_i^2 = 0$ for each index $...
Naysh's user avatar
  • 455
7 votes
2 answers
569 views

Deriving consequences of identities

Suppose we are given a variety in the universal algebra sense. For concreteness, suppose that we have two binary operations $+,\cdot$, three unary operations $-,\ast,'$, and two zeroary operations $0,...
Pace Nielsen's user avatar
1 vote
2 answers
322 views

Condition for equality of modules generated by columns of matrices

Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
Rahul Sarkar's user avatar

1
3 4
5
6 7
67