Strongly J-Clean Matrices Over Local Rings

Abstract

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8, Theorems 2.7 and 3.2], [9, Theorem 2.6], and [11, Theorem 7].