Strong cleanness of matrix rings over commutative rings

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 Mn(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. As in [10] a ring R is called clean if each element of R is the sum of an idempotent and a unit. In [8] Han and Nicholson proved that a ring R is clean if and only if Mn(R) is clean for every integer n ≥ 1. It is easy to check that each local ring is clean and consequently every matrix ring over a local ring is clean. On the other hand a ring R is called strongly clean if each element of R is the sum of an idempotent and a unit that commute. Recently, in [12], Chen and Wang gave an example of a commutative local ring R with M2(R) not strongly clean. This motivates the following interesting question: what are the commutative local rings R for which Mn(R) is strongly clean for each integer n ≥ 1? In [4], Chen, Yang and Zhou gave a complete characterization of commutative local rings R with M2(R) strongly clean. So, from their results and their examples, it is reasonable to conjecture that the Henselian rings are the only commutative local rings R with Mn(R) strongly clean for each integer n ≥ 1. In this note we give a partial answer to this problem. If R is Henselian then Mn(R) is strongly clean for each integer n ≥ 1 and the converse holds if R is an integrally closed domain, a valuation ring or if its residue class field is algebraically closed. All rings in this paper are associative with unity. By [11, Chapitre I] a commutative local ring R is said to be Henselian if each commutative module-finite R-algebra is a finite product of local rings. It was G. Azumaya ([1]) who first studied this property which was then developed by M. Nagata ([9]). The following theorem gives a new characterization of Henselian rings. Theorem 1. Let R be a commutative local ring. Then the following conditions are equivalent: (1) R is Henselian; (2) For each R-algebra A which is a direct limit of module-finite algebras and for each integer n ≥ 1, the matrix ring Mn(A) is strongly clean; (3) Each R-algebra A which is a direct limit of module-finite algebras is clean. 2000 Mathematics Subject Classification. Primary 13H99, 16U99.