Rings with fine idempotents

Abstract

An idempotent in a ring is called fine (see G. Călugăreanu and T. Y. Lam, Fine rings: A new class of simple rings, J. Algebra Appl. 15(9) (2016) 18) if it is a sum of a nilpotent and a unit. A ring is called an idempotent-fine ring (briefly, an [Formula: see text] ring) if all its nonzero idempotents are fine. In this paper, the properties of [Formula: see text] rings are studied. A notable result is proved: The diagonal idempotents [Formula: see text] ([Formula: see text]) are fine in the matrix ring [Formula: see text] for any unital ring [Formula: see text] and any positive integer [Formula: see text]. This yields many classes of rings over which matrix rings are [Formula: see text].

Decompositions of matrices into diagonalizable and square-zero matrices

“…Lam decomposition have high indices of nilpotence because they correspond to the strictly upper part of a matrix in good form. 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].…”

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

“…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].…”

“…M 9 (F)e(2,7,4) where e (2,7,4) = e 2,2 + e 7,7 + e 8,8 + e 9,9 . Thus, N C = N 1,4,4 + N 2,7,4 satisfies N 4 C = 0 andC + N C =…”