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.
3,342
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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?
...
4
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3
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467
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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
...
4
votes
3
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381
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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 ...
4
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2
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460
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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=...
4
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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 ...
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4
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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 ...
4
votes
1
answer
473
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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 $...
4
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2
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389
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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 $\mathrm{GL}_n(\mathbb C)$...
4
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2
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437
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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):
...
4
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1
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493
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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 ...
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3
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503
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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 ...
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4
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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, ...
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2
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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 ($A_n=0$ ...
4
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2
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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 $...
4
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488
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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\...
4
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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, ...
4
votes
2
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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 $\lambda_1,\dotsc,\lambda_n \in k$. ...
4
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1
answer
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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 ...
4
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1
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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 !?
4
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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 ...
4
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2
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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 ...
4
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2
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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 ...
4
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1
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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 ...
4
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1
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330
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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^...
4
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1
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360
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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^{-...
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2
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426
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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 $...
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2
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640
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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 ...
4
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1
answer
885
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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$ ...
4
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2
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452
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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] = ...
4
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2
answers
412
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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 ...
4
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3
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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 ...
4
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588
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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 ...
4
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1
answer
671
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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:
...
4
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3
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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 $\...
4
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2
answers
274
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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 ...
4
votes
2
answers
260
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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{...
4
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1
answer
241
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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 ...
4
votes
1
answer
281
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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 ...
4
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1
answer
399
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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
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2
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335
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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{...
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4
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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 ...
4
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2
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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 ...
4
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2
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1k
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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 ...
4
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1
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258
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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.
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1
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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: $G_iS$ =...
4
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2
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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 $D$, ...
4
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2
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334
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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 ...
4
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1
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293
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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 ...
4
votes
1
answer
479
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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 ...
4
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2
answers
171
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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 $D_I$ ...