Strong cleanness of the 2x2 matrix ring over a general local ring

A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [G. Borooah, AT Diesl, TJ. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (1) (2008) 281-296] completely characterized the commutative local rings R for which M(n) (R) is strongly clean. For a general local ring R and n > 1, however, it is unknown when the matrix ring M(n)(R) is strongly clean. Here we completely characterize the local rings R for which M(2)(R) is strongly clean. (C) 2008 Elsevier Inc. All rights reserved.