On Strongly Clean Matrix and Triangular Matrix Rings

Abstract

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson ( 1999 Nicholson , W. K. (1999). Strongly clean rings and Fitting's lemma. Comm. Algebra 27:3583–3592. [CSA] [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.