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Results tagged with ra.rings-and-algebras
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user 750
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.
5
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0
answers
115
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Ring properties which can be checked on sufficiently many Ore localizations
Let $R$ be a non-commutative Ore domain, and let $R[S_i^{-1}]$ be a finite set of Ore localizations. The absence of a geometric theory means there is no obvious candidate for when this set of localiz …
- 13k
18
votes
What is the "right" definition of a ring?
Much like abelian groups and groups, commutative rings and non-commutative rings have different motivations in my mind.
As people have said, non-commutative rings are naturally endomorphisms of abeli …
- 13k
3
votes
0
answers
163
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Pulling out factors in a Noetherian Domain
Let $R$ be a Noetherian domain (not-necessarily commutative), and let $S$ be a Noetherian subring of $R$. An element $r\in R$ is left $S$-irreducible if, for any $s\in S$ and $r' \in R$ with $sr'=r$, …
- 13k
17
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Can a module be an extension in two really different ways?
I believe this is a counter example. Let $R=\mathbb{C}[x]$, and consider finite-dimensional modules (ie, f.d. vector spaces equipped with a distinguished endomorphism). For convenience, I will iden …
- 13k
5
votes
Idempotents in Rings of Differential Operators
Here's a cute partial result, classifying idempotents of order 1.
Let $\delta$ be an idempotent differential operator of order 1. Then there is a unique decomposition $R\simeq A\oplus M$, with $A$ a …
- 13k
7
votes
1
answer
566
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Depth Zero Ideals in the Homogenized Weyl Algebra
Let $\mathcal{D}$ be the $n$th Weyl algebra $ \mathcal{D} :=k[x_1,...,x_n,\partial_1,...,\partial_n] $, where $\partial_ix_i-x_i\partial_i=1$.
Let $\widetilde{\mathcal{D}}$ be its Rees algebra, whi …
- 13k
26
votes
1
answer
990
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Idempotents in Rings of Differential Operators
Differential Operators on General Commutative Rings
Let k be an algebraically closed field of characteristic zero, and let R be a commutative k-algebra. Then a (Grothendieck) differential operator on …
- 13k
12
votes
1
answer
640
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Does the image of a differential operator always contain an ideal?
Let $\delta$ denote a non-zero complex algebraic differential operator in a single variable x. That is, it can be written as a sum
$$ \delta = \sum_i f_i\partial_x^i$$
where there $f_i$ are complex p …
- 13k
10
votes
Proof a Weyl Algebra isn't isomorphic to a matrix ring over a division ring
A different proof would be to show that a Weyl algebra is not semisimple, that is, that it is not a direct sum of simple submodules as a left module over itself. However, note that there is an infini …
- 13k