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.
1,088
questions with no upvoted or accepted answers
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Software for BMW algebra calculations?
Does software exist for computations in the BMW algebra?
For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...
- 291
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223
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Series in topological rings that only converge if almost all summands are zero
While trying to understand a certain topological ring better, I stumbled onto the following question.
Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a ...
- 9,549
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How to compute the abelianization of the representation theory of a Hopf algebra?
I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.
Version 1:
Let $(C,\otimes)$ be any monoidal ...
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130
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Does Mittag-Lefflerness descend?
I have read in the Stacks Project that if $A \to B$ is a faithfully flat ring homomorphism, $M$ is an $A$-module, and $M \otimes_A B$ is a flat, Mittag-Leffler $B$-module, then $M$ is a flat, Mittag-...
- 523
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164
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Classifying algebras with two idempotent generators and involution
Say that a finite-dimensional algebra $H$ over a field $K$ is dihedral if $H$ is generated by idempotents $P_1$ and $P_2$ and there is an algebra involution interchanging $P_1$ and $P_2$.
For example,...
- 1,001
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315
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Testing isomorphism of finitely generated algebras
Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and
$C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated $\...
- 7,521
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295
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Is there an idempotent measure on the free LD system?
This is a follow up question to MO question "Idempotent measures on the free binary system?".
Let $(A,*)$ be the free binary operation on one generator which satisfies the left self distributive law:
...
- 3,637
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145
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Decomposing an endomorphism as a tensor product
$\DeclareMathOperator\End{End}$Let $f$ be an endomorphism of the finite-dimensional vector space $V$, over the field $K$. The question of whether $f$ is decomposable, that is, whether $V$ can be ...
- 2,195
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A standard name for the algebraic structure on a projective line?
Question: Is there any name for the natural algebraic structure of the projective line?
Algebraically, a projective line over a field is a set $L$ endowed with two binary operations $+$ and $\cdot$ ...
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182
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From group ring to ring ring?
For a group $G$, the set $\mathbb{Z}[G]$ of all formal $\mathbb{Z}$ linear combinations is a ring with unit. Now the set $\mathbb{Z}[\mathbb{Z}[G]]$ gets the structure of a ring from the addition in $\...
- 119
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Connections in non-commutative geometry
Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...
- 401
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Rings where all indecomposable modules are projective or injective
Let $A$ be a semi-perfect noetherian ring.
Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective?
Im also interested in ...
- 26.1k
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275
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Serre subcategories of the category of chain complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R$ be a commutative $k$-algebra.
We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...
- 639
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174
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The precise relationship between (moduli space of) finite-dimensional commutative local $\kappa$-algebras and number theory?
In [BP08] Poonen constructs and studies $\mathfrak{B}_n$, the moduli space of all based $n$-dimensional commutative associative unital $\kappa$-algebras, where $\kappa$ is an algebraically closed ...
- 6,730
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220
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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-...
- 1,746
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175
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Bar constructions and pushouts
Suppose that $\mathsf S$ is a span of associative algebras (or, more generally, if you'd like, any type of object admitting a bar-cobar formalism) and let $A$ be its pushout.
Is there any hope of ...
- 1,534
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Reference for countable and uncountable algebraic closures of $\mathbb{Q}$ in ZF
The following facts seem to be part of the folklore (where $\mathsf{ZF}$ means Zermelo-Fraenkel set theory with no axiom of choice):
it is consistent with $\mathsf{ZF}$ that there exists an ...
- 30.2k
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106
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Indecomposable objects in iterated functor categories
Let $\mathcal{O}$ be a DVR with a uniformizer $\pi$ and a finite quotient field $\mathbb{F}_q=\mathcal{O}/\pi$. Write $\mathcal{O}_r = \mathcal{O}/(\pi^r)$. Fix now an integer $r\geq 2$. We define ...
- 4,979
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Non-commutative rings where every non-unit is contained in a completely prime ideal
Below, all rings are associative and unital; and the word "ideal" always refers to a two-sided ideal.
Let's stipulate that a ring $R$ has property (P) if every non-unit of $R$ is contained ...
- 9,636
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90
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Structure of finitely generated $\mathbb{Z}/p^n\mathbb{Z}[[S,T]]$-modules
Let $\Omega=\mathbb{Z}/p\mathbb{Z}[[S,T]]$. $\Omega$ is a commutative, Noetherian and integrally closed domain of Krull dimension 2. According to Bourbaki's commutative algebra VII $\S 4$, if $M$ is a ...
- 409
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Rings such that torsion-free/flat/projective modules are flat/projective/free
While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
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158
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Representation theory terminology question
For a paper I'm writing, I need a term for a representation-theoretic concept that I'm sure someone has thought of before, so I thought I'd ask here rather than just make something up.
Let $G$ be a ...
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406
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A lemma concerning conjugations and normal subgroups (related to a theorem of Frobenius)
In the paper "On A Theorem of Frobenius" written in 1969, Prof. Richard Brauer, for the first time, presented a character-free proof to the Frobenius theorem (i.e. counting the number of ...
- 51
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130
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Can a braided fusion category have an order-2 Morita equivalence class which cannot be simultaneously connected and isomorphic to its opposite?
Let $\mathcal{B}$ be a braided fusion category over $\mathbb{C}$. Let me write $\mathrm{Alg}(\mathcal{B})$ for the set of isomorphism classes of unital associative algebra objects in $\mathcal{B}$, ...
- 53.1k
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240
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Growth rate of cohomology
Fix a finite dimensional graded vector space $V$, a differential $d$ on $V \otimes V $ $(i.e.\,\, d^2=0\,\, and \,\, deg\, d =1)$ such that $(d_{1,2} \otimes id_3)$ anti-commutes with $((-1)^{deg}_1 \...
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One-parameter family of algebra structures: characterizing trivial deformation as adjoint 2-cocycle being a 2-coboundary
$\DeclareMathOperator\C{\mathbf{C}}$Motivation: this post discusses a simple criterion for a 1-parameter family of $n$-dimensional complex algebras to be a "trivial deformation", i.e., be ...
- 60.4k
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100
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Center of symplectic derivation Lie algebra
Morita–Sakasai–Suzuki studied the graded Lie algebra $\mathfrak{h}_{g,1}$ of symplectic derivations, as well as variations $\mathfrak{h}_{g,\ast}$ and $\mathfrak{h}_g$. This is the Lie algebra of ...
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How does a map from permutahedra to associahedra factor through multiplihedra?
Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $...
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73
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Reference request: associative subalgebras of Cayley algebras are at most 4-dimensional
By a Cayley algebra I mean an 8-dimensional algebra (over an arbitrary field) formed in the Cayley-Dickson process. (They are also called octonion algebras, but I prefer to reserve the term octonion ...
- 2,437
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Is there a list of all real unital subalgebras of M(2,C)?
Is there a complete classification of all real unital subalgebras of $M(2,\mathbb C)$ up to isomorphism? The list should include $M(2,\mathbb C)$, the quaternions, complex numbers, split-complex ...
- 4,843
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311
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A fusion ring identity
Fusion rings
I'll more or less stick to the presentation given in this question: [1]
We define a fusion ring as follows: consider a free $\mathbb{Z}$-module $\mathbb{Z}\mathcal{B}$ with finite basis ...
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elementary matrices over a regular ring
Let $n\in\mathbb{N}$ with $n\geq 3$. Let $A$ be a regular ring and $\gamma\in GL_{n}(A)$. We suppose that $\gamma$ is homotopic to the identity, i.e. there exists $\alpha\in GL_{n}(A[X])$ such that $\...
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Finiteness questions for enveloping algebras
Let $\mathfrak{g}$ be a Lie algebra over a field of characteristic $0$. The interesting case for these questions is when $\mathfrak{g}$ is infinite dimensional.
Is $U(\mathfrak{g})$ a coherent ...
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140
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Open problems about Morita and derived invariants
Are there properties of rings of which one does not know whether they are Morita or derived invariances?
For a recent such example for Morita invariance, see https://www.sciencedirect.com/science/...
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Any f.p. faithful simple module over a primitive group ring?
Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$.
There ...
- 1,528
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Embedding the Mészáros subdivision algebra in an Orlik-Terao localization
The following is an open question (Question 4.1) from my paper $t$-Unique
Reductions for Mészáros's Subdivision Algebra (published version in
SIGMA 2018, and slightly updated preprint
version with ...
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474
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Slightly noncommutative Nakayama's lemma?
Nakayama's lemma asserts the following. If $R $ is a commutative ring with an element $s$, and $M$ is a finitely generated module such that $sM = M$, then there exists $r \in R$ such that $rM =0$ and $...
- 4,669
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Jacobson radical of a tensor product
Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...
- 6,174
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von Neumann regular ring homomorphisms
Let us call a ring homomorphism $f\colon R\rightarrow S$ von Neuman regular if it has the property that for every left $S$-module $M$, the left $R$-module $f^*M$ is flat.
In particular, $\mathrm{id}...
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Are these element in a group algebra of a torsion-free group zero divisors?
Let $G$ be an arbitrary torsion-free group. For $x,y\in G$, which of these elements can be decided immediately not to be zero divisors in $\mathbb ZG$ (or in $\mathbb CG$)?
$$1+x+y,\quad 4+x+x^{-1}+y+...
- 2,118
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214
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Constructing a noncommutative algebra from a commutative algebra
I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
- 111
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112
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The cyclic modules and self injective ring
It is well-known that if R is a Noetherian ring, and every finitely generated right R-module embeds in projective, then R is a self-injective.
My question is that could one replace "finitely ...
- 107
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182
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Which rings are the endomorphisms ring of some abelian groups?
Which rings are (isomorphic to) the endomorphisms ring of some abelian group? Is there any necessary and sufficient condition?
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Finitely generated submodules of projectives lie inside f. g. projectives?
Let $R$ be a (not necessarily commutative) ring.
If $M$ is a finitely generated submodule of a projective module $P$, is there a finitely generated projective submodule $P'$ such that
$M \subseteq P'...
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185
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$p$-adic valuation in the ring $\mathbb{Z}/p^k\mathbb{Z}$
Assume $p$ is a prime number, $M$ be a non-negative integer and denote by $(\mathbb{Z}/p^M\mathbb{Z})^*$ the units of $\mathbb{Z}/p^M\mathbb{Z}$. Now consider the partition of $\mathbb{Z}/p^M\mathbb{Z}...
- 2,949
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Real endomorphism algebra of abelian surface is never $\mathbb{C}$?
I'm reading about the Sato Tate conjecture for genus 2, and I came to the paper here. This breaks the conjecture into 6 different parts, based on the real endomorphism algebra of the surface, or the ...
- 171
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138
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Chains of right annihilators in group rings
See the update below
This problem emanates from a question on not-so-simple random walks on finitely generated groups. But to explain the connections would require an extremely long essay.
Let $G$ be ...
- 4,669
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315
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Is there a matrix with this specific quadratic determinant?
We have $\det M=(a+b)(c+d)$ where
$M=\begin{bmatrix}
a& 0& -1& 0\\
0& c& 0& -1\\
b& 0& 1& 0\\
0& d& 0& 1
\end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
- 13.7k
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245
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What is the relationship between Frobenius extensions and Separable extensions
Let $R\to S$ be an extension of possibly non-commutative rings. I am interested in the relationship between $R\to S$ being Frobenius and it being separable.
If it is a Frobenius extension, then there ...
- 9,549
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114
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A general form of a maximal totally isotropic subspace in the split octonion algebra
Let $\mathbb O'$ be the split octonion algebra over $\mathbb R$. For each nonzero divisor of zero $x\in \mathbb O'$ $\mbox{($x \neq 0, N(x)=0$)}$ the kernel of the left multiplication by $x$, $Ker ...
- 175