Strong cleanness of matrix rings over commutative rings

Details Strong cleanness of matrix rings over commutative rings Strong cleanness of matrix rings over commutative rings

2008-04-08

arxiv.org/abs/0804.1221v1

Communications in Algebra 36, 2 (2008) 346-351

Mathematics

Rings and Algebras

Scientific paper

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.

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