A Note on Commutative Nil-Clean Corners in Unital Rings

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-C\v{a}lug\v{a}reanu-Danchev-Micu in Lin. Algebra \& Appl. (2013) that if $R$ is a commutative nil-clean ring, then the full matrix ring $\mathbbm{M}_n(R)$ 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.