All Questions
Tagged with local-rings noncommutative-rings
7 questions
5
votes
1
answer
191
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Are module finite algebras over semiperfect rings again semiperfect?
Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
- 35.2k
2
votes
1
answer
112
views
Local rings whose the endomorphism rings of E(R/J) is division ring
Let $R$ be a local ring with maximal ideal $J$. Assume that ${\rm End}_{R}({\rm E}(R/J))$ is a division ring (${\rm E}(R/J)$ means the injective envelope of $R/J$). Does $R/J$ is injective?
- 1,509
6
votes
1
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443
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Ring in which $x^n-x$ is central for every $x$
Let $R$ be a ring , $n \gt 1$, such that for all $x \in R$: $x^n-x \in Z(R)$, the center of $R$. Does it follow that $R$ is commutative?
For $n=2,3$ this is pretty straightforward to prove. But what ...
- 16.2k
1
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1
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147
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Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not ...
- 2,928
0
votes
1
answer
248
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Given a unitary commutative ring $R$, what are the rings $R\langle x,y\rangle/(x^2-A,y^2-B,yx-a-bx-cy-dxy)$ called
We are studying the rings
$$
R \langle x, \, y \rangle\,\big/\left(x^2-A, \, y^2-B, \, yx-a-bx-cy-dxy \right)
$$
Do you know if they have a name?
- 4,991
6
votes
1
answer
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Do you know which is the minimal local ring that is not isomorphic to its opposite?
The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
- 26.5k
16
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2
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A graded ring $R$ is graded-local iff $R_0$ is a local ring?
I asked this question some months ago on math.stackexchange.com:
https://math.stackexchange.com/questions/126810/a-graded-ring-r-is-graded-local-iff-r-0-is-a-local-ring
It would be great (for me) to ...
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