Strong cleanness of matrix rings over commutative rings

Francois Couchot (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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.

Mots clés: clean ring; strongly clean ring; local ring; Henselian ring; matrix ring; valuation ring

Domaines: Anneaux et algèbres [math.RA]

Francois Couchot. Strong cleanness of matrix rings over commutative rings. Communications in Algebra, 2008, 36 (2), pp.346-351. ⟨hal-00271101⟩