On 2-nil-good rings

Abstract

A ring [Formula: see text] is defined to be 2-nil-good if every element in [Formula: see text] is the sum of two units and a nilpotent. Fundamental properties of such rings are obtained. We prove that every strongly [Formula: see text]-regular ring is 2-nil-good if and only if the identity is the sum of two units. One of the main result of this paper is that every square matrix ring over J-fine rings is 2-nil-good. We establish 2-nil-good property for Morita contexts. This implies, in particular, that every matrix ring over 2-nil-good rings is 2-nil-good. Furthermore, we prove that the ring [Formula: see text] of all lower triangular diagonal-finite matrices over a 2-good ring [Formula: see text] is 2-nil-good.