Strong cleanness of the $2\times 2$ matrix ring over a general local ring

Abstract

A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the commutative local rings $R$ for which ${\mathbb M}_n(R)$ is strongly clean. For a general local ring $R$ and $n>1$, however, it is unknown when the matrix ring ${\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local rings $R$ for which ${\mathbb M}_2(R)$ is strongly clean.