Rings with fine idempotents.

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 I F ring) if all its nonzero idempotents are fine. In this paper, the properties of I F rings are studied. A notable result is proved: The diagonal idempotents E 1 1 + ⋯ + E i i (i = 1 , ... , n) are fine in the matrix ring ℳ n (R) for any unital ring R and any positive integer n. This yields many classes of rings over which matrix rings are I F. [ABSTRACT FROM AUTHOR]