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.

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Higher categories and semirings

Maybe my thinking here is completely wrong headed, but this seems like something that ought to have been answered before. Here is my question: What is the (n - )categorical analogue of a semiring? ...
Mikola's user avatar
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Computer algebra systems that can handle real-closed fields?

I'm trying to solve a big system of: quadratic equations with coefficients in $\mathbb{Z}$, each in 6 variables, quadratic inequalities with coefficients in $\mathbb{Z}$, each in 6 variables, and ...
Andrey Mishchenko's user avatar
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3 answers
381 views

Intuition for left Hopf-modules

I'm reading Radford's text on Hopf Algebras right now and I'm a bit confused about the definition of a left $A$-Hopf algebra. The definition given in the book is: Let $A$ be a $\Bbbk$-bialgebra. A ...
Sunbeam's user avatar
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Does a nonabelian Picard group exist?

Over a noncommutative algebra $A$ we have no problem in defining invertible bimodules (as in the book by Bass on algebraic $K$-theory) - corresponding to line bundles over topological spaces $X$ if $A=...
Edwin Beggs's user avatar
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1 answer
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Is it true that Nature promotes products?

I hope this question is not unreasonable. We all know how to take products of numbers, this generalises to a huge amount of different types of products in mathematics. In a certain sense this notion ...
aglearner's user avatar
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Is there an analogue of finite fields for products of two prime powers?

The collection of prime powers can be characterized in the following way: There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique ...
Alan Haynes's user avatar
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Are eigenvalues preserved under derived equivalence?

Let $A$ and $B$ be finite dimensional algebras such that $A$ and $B$ are derived equivalent. Denote by $C_A$ (resp. $C_B$) the Cartan matrix of $A$ (resp. $B$). Then does the set of eigenvalues of $...
Sola.322's user avatar
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Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$

For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x_1,...,x_n)$. For a subgroup $G$ of $\mathrm{GL}_n(\mathbb C)$...
user521337's user avatar
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437 views

Polynomial ring operations on $\mathbb{Z}$

I have asked this on Math Stack Exchange but without answers: The usual ring operations on $\mathbb{Z}$ can be defined via polynomials in $\mathbb{Z}[a,b]$ (when viewing $a,b$ as variables): ...
Martin's user avatar
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Division ring on a field

Suppose that $F$ is a field. Show that there exists a $F$-division algebra $D$ with two elements $a\neq b\in D$ such that $a^2-2ab+b^2=0$. In the field extensions we know that $a^2-2ab+b^2=0$ if and ...
MH.Fakharan's user avatar
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Non-principal prime ideals in infinite distributive lattices

Given an infinite distributive lattice $L$, does $L$ contain a non-principal prime ideal $I$, or a non-principal prime filter $F$? ($I$ is said to be principal if there is $x\in L$ such that $I=\{y\in ...
Dominic van der Zypen's user avatar
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4 answers
381 views

What is an ideal-supporting algebra?

I'm sorry if this question is too elementary, but I asked it at MathStackExchange and it received no responses. On the Wikipedia page for congruence relation it mentions how for groups and rings, ...
Taliberius 4's user avatar
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2 answers
415 views

Are homogeneous components of f.g graded modules f.g ?

Suppose $M= \bigoplus_{n\in \mathbb Z} M_n$ is a finitely generated graded module over a Noetherian graded commutative ring $A=\bigoplus_{n\in \mathbb Z}A_n$. If $A$ is positively graded ($A_n=0$ ...
tj_'s user avatar
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When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau?

I would like to ask a simple question. Let $A=\mathbb{C}\langle x_{1},\dots,x_{n} \rangle/I$, where $I$ is the two-sided ideal generated by $x_{i}x_{j}=a_{ij}x_{j}x_{i}$ for $1\le i,j\le n$. We say a $...
Muon's user avatar
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Is the functor of divided powers a weakly monoidal functor?

Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every $m\...
Ivan Yudin's user avatar
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1 answer
341 views

How to have MAGMA work with subgroup of ATLASGroups?

I'm trying to work with various maximal subgroups of the Thompson sporadic group. The command Group("Th"); which works for some of the sporadic groups, ...
NewViewsMath's user avatar
4 votes
2 answers
277 views

Free augmented algebras

What is the "correct" definition of a free augmented commutative algebra? At least two definitions come to my mind: Fix a commutative ring $k$. We need elements $\lambda_1,\dotsc,\lambda_n \in k$. ...
Martin Brandenburg's user avatar
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1 answer
1k views

Is the universal enveloping algebra of a finite-dimensional Lie algebra (left) noetherian?

The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is a flat deformation of $S(\mathfrak{g})$, so these algebras should be similar in many ways. Does at least this general similarity ...
Oleg's user avatar
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1 answer
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A Property of Finite Rings

Consider a finite ring $R$ with identity. If every left ideal of $R$ is two-sided then is it true that any right ideal of $R$ is also two-sided !?
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Is the normalisation of an integral noetherien dimension one ring a finite morphism?

This feels like something I should know but I can't find an answer in Liu or in Atiyah-MacDonald, or a counter-example. To state the question again: let $A$ be an integral Noetherien ring of Krull ...
name's user avatar
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4 votes
2 answers
865 views

What is the correct formulation of the CDE triangle?

The CDE triangle with C Cartan matrix and D decomposition matrix is well-known for finite groups. I have not seen a full account of this for finite dimensional algebras which I find surprising as ...
Bruce Westbury's user avatar
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2 answers
566 views

A subalgebra of a Frobenius algebra that is not again a Frobenius algebra?

A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. (See here for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a ...
Didier de Montblazon's user avatar
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1 answer
298 views

What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$?

I'm now interested in the modular representation of symmetric groups. It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S_{n}$ over a field ...
gualterio's user avatar
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1 answer
330 views

Invertible bimodules and projectivity

Let $A$ be a noncommutative algebra over a field, say $\mathbb{C}$ or $\mathbb{R}$, and let $L$ be a bimodule over $A$. If $L$ is invertible, that is, if the dual right $A$-module $L^*$ satisfies $$ L^...
Rodrigo Alfonso de la Paz's user avatar
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1 answer
360 views

Diophantine equation in Laurent polynomials

(This is a modified repost of a question from MSE; since it came out of research, I thought it might be appropriate to post it here.) Consider the equation \begin{equation*} P(x, x^{-1})^m + Q(x, x^{-...
user137's user avatar
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426 views

Global dimension of the tensor algebra

Let $R$ be a semisimple ring with a non-zero $R$-bimodule V. Let $T_R(V):= \bigoplus\limits_{k=0}^{\infty}{V^{\otimes_k}}$ be the tensor algebra of $V$. Question 1: Is there a simple proof that $...
Mare's user avatar
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640 views

Modules over infinite rings which can not be a finite union of their proper submodules

It is well known that a vector space over an infinite field cannot be a finite union of its proper subspaces. Does this fact have an immediate and obvious generalization to modules over infinite ...
Ali Taghavi's user avatar
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1 answer
885 views

Regular functions on a product of varieties

Let $k$ be an algebraically closed field and let $X$, $Y$ be varieties over $k$. Let us denote by $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ the $k$-algebra of regular functions on $X$ and $Y$ ...
Anonymous's user avatar
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2 answers
452 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] = ...
Alan's user avatar
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4 votes
2 answers
412 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 ...
Cusp's user avatar
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4 votes
3 answers
793 views

smooth connected affine scheme over Z has good reduction almost everywhere

Let $f(x_1,\ldots,x_n)\in\mathbf{Z}[x_1,\ldots,x_n]$ be a polynomial. Assume that the variety cut out by $f$ is smooth and connected (so irreducible) over $\overline{\mathbf{Q}}$. Where can I find a ...
Hugo Chapdelaine's user avatar
4 votes
1 answer
588 views

chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,...,x_n,...]$

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true? Question: Is there any maximal ...
Alex's user avatar
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4 votes
1 answer
671 views

Are all (commutative) rngs ideals of (commutative) rings? [closed]

To avoid repeating it endlessly, assume all rings and rngs are commutative. I do not know if this is necessary. The question then is exactly the title, but I think a stronger statement is true: ...
Richard Rast's user avatar
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4 votes
3 answers
1k views

Lattice of subcategories: subobject classifier in Cat

Two short questions: Is there any work classifying the lattice of subcategories of an arbitrary (sufficiently small) category $\mathcal{C}$, similar to the way that the set of subsets of set $\...
supercooldave's user avatar
4 votes
2 answers
274 views

Relation of the first Hochschild cohomology and the outer automorphism group

Let $R$ be a ring. Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite? (It is not true, by the two answers. Is it ...
Mare's user avatar
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4 votes
2 answers
260 views

Embedding of a division algebra into a matrix algebra over its centre

Let $K$ be a number field and let $D$ be a central division algebra over $K$. Let $d$ be the index so that $[D:K]=d^2$. What is the minimal $n$ such that there exists an embedding of $D$ into $\mathrm{...
Henri Johnston's user avatar
4 votes
1 answer
241 views

Left- (right-) multiplications of an algebra that are derivations

Let us say that $A$ is a (finite-dimensional) algebra over a field of characteristic zero. We can assume commutativity but not associativity, if that makes it easier. Indeed, I am mostly interested in ...
Claudio Gorodski's user avatar
4 votes
1 answer
281 views

When is an algebra defined by generators and relations finite-dimensional and satisfies Poincaré duality?

When is an algebra defined by generators and relations finite-dimensional and satisfies Poincaré duality? EDIT: My question is not very concrete. Rather I am wondering if there is anything known in ...
asv's user avatar
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4 votes
1 answer
399 views

Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$

Edit: According to the comment of Todd Trimble, I revise the question. What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...
4 votes
2 answers
335 views

When is $\Omega^1$ an equivalence?

Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules. When is the functor $\Omega^1 : \underline{...
Mare's user avatar
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4 votes
4 answers
2k views

Products of Boolean algebras and probability measures thereon

These are really two questions, but the second presupposes the first. First, let $( B_i )_{i\in I}$ be an arbitrary family of Boolean algebras. I want to directly form a product of them that is like ...
Alexander Pruss's user avatar
4 votes
2 answers
733 views

Constructing a ring from an abelian group in a minimal way

I'm looking for a specific construction, taking an abelian group (with designated element) $(G,+,1)$ to a commutative ring $(R,+,\cdot,1)$, where $G\subset R$ as a pointed abelian group, and which is ...
Richard Rast's user avatar
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4 votes
2 answers
1k views

Semiprime (but not prime) ring whose center is a domain

The center of a prime ring is a domain and the center of a semiprime ring is reduced. Now I have no evidence to believe that if the center of a semiprime ring R is a domain, then R has to be a ...
carlos's user avatar
  • 279
4 votes
1 answer
258 views

Existence of module with periodic resolution

Let $R$ be a Gorenstein local ring. Does there always exist a module $M$ having eventually periodic minimal free resolution? Any reference is also appreciated.
SKS's user avatar
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4 votes
1 answer
258 views

Calculating the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$

I want to calculate the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$ when its generators $G_i$ (whose number is finite) are known. For any matrix $S$ that commutes with the group: $G_iS$ =...
Jim's user avatar
  • 215
4 votes
2 answers
181 views

Ehrlich's Characterization of Unit Regular Elements

I recently came across the following characterization of unit regular elements of an endomorphism ring (Corollary to Theorem 1 in this article). Let $M$ be a vector space over the division ring $D$, ...
dbossaller's user avatar
4 votes
2 answers
334 views

Are there examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras?

Are there any examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras? Or can one show that none exist? I went through this list of all complex associative ...
Andi Bauer's user avatar
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4 votes
1 answer
293 views

Can base-change be non-surjective on Brauer groups?

Is there a finite-degree separable field extension $\mathbb{K} \subset \mathbb{L}$ such that the induced map on Brauer groups $\operatorname{Br}(\mathbb{K}) \to \operatorname{Br}(\mathbb{L})$ is not a ...
Theo Johnson-Freyd's user avatar
4 votes
1 answer
479 views

List of Casimir elements of low dimensional Lie algebras

I am interested in explicit formulae for the Casimir elements (or "Casimir operators") of low-dimensional, real, non-Abelian Lie algebras (d=2,3, and possibly 4). I am wondering if there is any ...
Aureliano Tinajero's user avatar
4 votes
2 answers
171 views

Covering derivations of a quotient algebra

Let $(\mathcal{A},+,·)$ an algebra and $\mathcal{I}$ an ideal of $\mathcal{A}$. Is easy to check that if $D\in Der(\mathcal{A})$ with $D(\mathcal{I})\subseteq I$ then $D$ induces a derivation $D_I$ ...
mct_brasil's user avatar

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