Geodesic
A note on commutative nil-clean corners in unital rings
The Bulletin of Irkutsk State University. Series Mathematics, Tome 29 (2019), pp. 3-9
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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. 
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     title = {A note on commutative nil-clean corners in unital rings},
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P. V. Danchev. A note on commutative nil-clean corners in unital rings. The Bulletin of Irkutsk State University. Series Mathematics, Tome 29 (2019), pp. 3-9. http://geodesic.mathdoc.fr/item/IIGUM_2019_29_a0/

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