All Questions
5 questions
6
votes
1
answer
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 ...
- 365
6
votes
1
answer
499
views
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.
5
votes
1
answer
191
views
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 ...
- 412
1
vote
1
answer
147
views
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 ...
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?