Strong Cleanness of Matrix Rings Over Commutative 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 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 π-regular commutative ring.

Key Words:

• Clean ring
• Henselian ring
• Local ring
• Matrix ring
• Strongly clean ring
• Valuation ring

2000 Mathematics Subject Classification:

• Primary 13H99, 16U99

Notes

Communicated by R. Wisbauer.