Strong cleanness of matrix rings over commutative rings

TL;DR

This paper characterizes when matrix rings over commutative local rings are strongly clean, linking it to the Henselian property of the base ring and exploring conditions under which the converse holds.

Contribution

It establishes a new characterization of Henselian rings via strong cleanness of matrix rings and extends the analysis to various classes of commutative rings.

Findings

Matrix rings over Henselian rings are strongly clean.

The converse holds if the residue field is algebraically closed, or the ring is integrally closed or a valuation ring.

Strong cleanness extends to locally direct limit algebras over π-regular rings.

Abstract

Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $\mathbb{M}_n(R)$ is strongly clean for each integer $n>0$ if $R$ is Henselian and we show that the converse holds if either the residue class field of $R$ is algebraically closed or $R$ is an integrally closed domain or $R$ is a valuation ring. It is also shown that each $R$-algebra which is locally a direct limit of module-finite algebras, is strongly clean if $R$ is a $\pi$-regular commutative ring.