When is every matrix over a division ring a sum of an idempotent and a nilpotent?

Abstract

A ring is called nil-clean if each of its elements is a sum of an idempotent and a nilpotent. In response to a question of S. Breaz et al. in [1], we prove that the n×n matrix ring over a division ring D is a nil-clean ring if and only if D≅F2. As consequences, it is shown that the n×n matrix ring over a strongly regular ring R is a nil-clean ring if and only if R is a Boolean ring, and that a semilocal ring R is nil-clean if and only if its Jacobson radical J(R) is nil and R/J(R) is a direct product of matrix rings over F2.