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.
3,333
questions
4
votes
1
answer
226
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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}(...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
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. ...
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 (...
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 ...
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 ...
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,...
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}...
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 $B(2,3)$. ...
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 ...
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 $n$-...
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, ...
-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 $h_1, \dots, h_m$...
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 $n$-...
-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 ...
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 ...
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
.
...
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 ...
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 $...
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-...
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-...
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 ...
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, ...
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 (...
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)
...
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 ...
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 \...
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 $\mathfrak{p} \subset A$ , ...
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
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 ...
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 $A_F$-...
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 ...
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 $...
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-...
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 ...
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 ...
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 \...
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}\...
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 $...
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 $...
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,...
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 ...