|
Algebraic and logical methods in computer science and artificial intelligence
A note on commutative nil-clean corners in unital rings
P. V. Danchev Institute of Mathematics and Informatics of Bulgarian Academy of Sciences, Sofia,
Bulgaria
Abstract:
We shall prove that if $R$ is a ring with a family of orthogonal idempotents $\{e_i\}_{i=1}^n$ having sum $1$ such that each corner subring $e_iRe_i$ is commutative nil-clean, then $R$ is too nil-clean, by showing that this assertion is actually equivalent to the statement established by Breaz-Cǎlugǎreanu-Danchev-Micu in Lin. Algebra & Appl. (2013) that if $R$ is a commutative nil-clean ring, then the full matrix ring $\mathbb{M}_n(R)$ is also nil-clean for any size $n$. Likewise, the present proof somewhat supplies our recent result in Bull. Iran. Math. Soc. (2018) concerning strongly nil-clean corner rings as well as it gives a new strategy for further developments of the investigated theme.
Keywords:
nil-clean rings, nilpotents, idempotents, corners.
Received: 01.08.2019
Citation:
P. V. Danchev, “A note on commutative nil-clean corners in unital rings”, Bulletin of Irkutsk State University. Series Mathematics, 29 (2019), 3–9
Linking options:
https://www.mathnet.ru/eng/iigum379 https://www.mathnet.ru/eng/iigum/v29/p3
|
Statistics & downloads: |
Abstract page: | 132 | Full-text PDF : | 84 | References: | 12 |
|