When is the 2×2 matrix ring over a commutative local ring strongly clean?

Abstract

A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. Local rings are strongly clean. It is unknown when a matrix ring is strongly clean. However it is known from [J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, preprint, 2005] that for any prime number p, the 2 × 2 matrix ring M 2 ( Z p ) is strongly clean where Z p is the ring of p-adic integers, but M 2 (Z (p) ) is not strongly clean where Z (p) is the localization of Z at the prime ideal generated by p. Let R be a commutative local ring. A criterion in terms of solvability of a simple quadratic equation in R is obtained for M 2 (R) to be strongly clean. As consequences, M 2 (R) is strongly clean iff M 2 (RJxK) is strongly clean iff M 2 (R[x]/(x n )) is strongly clean iff M 2 (RC 2 ) is strongly clean.