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Results tagged with ra.rings-and-algebras
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user 78
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
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1
answer
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Normed algebras of indefinite signature?
Hurwitz's theorem states that a real possibly-non-associative algebra (meaning just a real vector space $V$ equipped with a map $m: V \otimes V \to V$) along with a positive definite quadratic form $| …
- 54.6k
6
votes
0
answers
86
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Can an algebra be isomorphic to its own algebra of $n^2 \times n^2$ matrices but not its own...
Is there an associative unital algebra $A$ which is isomorphic to its own algebra of $n^2\times n^2$ matrices $\operatorname{Mat}_{n^2}(A)$, but not isomorphic to its algebra of $n \times n$ matrices …
- 54.6k
2
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Can one bound the minimal degree of a solution to an inhomogeneous linear equation?
Suppose $A = \cup_{i\in \mathbb N} A_{\leq i}$ is a filtered algebra and $M = \cup_{i\in \mathbb N} M_{\leq i}$ a filtered $A$-module. In my case, I know $A$ to be Noetherian and $M$ to be finitely ge …
- 54.6k
4
votes
2
answers
322
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A hands-on description of a "completion" of the free commutative monoid on countably many ge...
This is basically an I'm-weak-at-algebraic-geometry question. I asked it as a warm-up question here, but Ilya N asked me to break that post up into several questions.
Consider the free commutative m …
- 54.6k
4
votes
1
answer
311
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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 …
- 54.6k
8
votes
1
answer
244
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Are annihilation modules in the quantum torus necessarily principal?
I hope that my question yields some standard fact from (noncommutative) ring theory. In discussions with other graduate students, we have outlined some approaches to tackling the question, but haven' …
- 54.6k
5
votes
0
answers
152
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What is the composition in SesquiAlg?
To motivate my question, I will describe a famous 2-category. First, there is the 1-category $\text{Vect}$ of vector spaces (over some fixed field). This category has a tensor product $\otimes$, and …
- 54.6k
12
votes
0
answers
275
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How can you unitalize a higher category?
Given an associative nonunital algebra $A$, there are (at least) two standard ways to produce a unital algebra $A'$ together with a map $A \to A'$. Following the discussion in the comments below, thes …
- 54.6k
13
votes
0
answers
338
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When does Hochschild homology commute with infinite products?
Let $A$ be an associative algebra. Its zeroth Hochschild homology $\mathrm{HH}_0(A)$ is the cokernel of the linear map $A^{\wedge 2} \to A$, $a \wedge b \mapsto ab - ba$. I.e. you quotient the vecto …
- 54.6k
16
votes
6
answers
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What is an algebraic group over a noncommutative ring?
Let $R$ be a (noncommutative) ring. (For me, the words "ring" and "algebra" are isomorphic, and all rings are associative with unit, and usually noncommutative.) Then I think I know what "linear alg …
- 54.6k
9
votes
2
answers
2k
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What are examples of cogenerators in R-mod?
Fill in the blank, please :)
Let $\mathcal C$ be a complete and cocomplete abelian category. A generator in $\mathcal C$ is an object $X \in \mathcal C$ such that every object $Y \in \mathcal C$ …
- 54.6k
9
votes
2
answers
805
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Strategies for proving a category is Noetherian?
Let $C$ be a small linear category over a commutative ring $R$. A representation of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation $[ …
- 54.6k
15
votes
3
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Which is the correct universal enveloping algebra in positive characteristic?
This is an extension of this question about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers.
Let $\math …
- 54.6k
20
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1
answer
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Reference request: Morita bicategory
I have two closely related questions:
Who first recognized that Morita equivalence was equivalence in the bicategory of Rings, Bimodules, and Intertwiners?
I've heard this bicategory called the "Mor …
- 54.6k
13
votes
1
answer
1k
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Why not _co_free modules?
Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and cocont …
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