Rings with fine nilpotents

Decompositions of matrices into a sum of invertible matrices and matrices of fixed nilpotence

“…The condition r ≤ t(k − 1) holds. As in the proof of Proposition 2.3, we use the classical division algorithm to express n − t = c(k − 1) + d for c = 1 and d = 3; hence, we consider the nilpotent matrices 3,5) ,…”

Decompositions of matrices into diagonalizable and square-zero matrices

“…The rings whose nonzero idempotents are fine turned out to be an interesting class of indecomposable rings and were studied in [2] by Cǎlugǎreanu and Zhou. In 2021, the same authors focused on rings in which every nonzero nilpotent element is fine, which they called N F rings, and showed that for a commutative ring R and n ≥ 2, the matrix ring M n (R) is N F if and only if R is a field; see [3].…”