A NOTE ON STRONGLY CLEAN MATRIX RINGS

Let R be an associative ring with identity. An element a. R is called strongly clean if a = e + u with e(2) = e is an element of R, u a unit of R, and eu = ue. A ring R is called strongly clean if every element of R is strongly clean. Strongly clean rings were introduced by Nicholson [7]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when R is local or strongly pi-regular. In this note, necessary conditions for the matrix ring IMn(R) (n > 1) over an arbitrary ring R to be strongly clean are given, and the strongly clean property of IM2 (RC2) over the group ring RC2 with R local is obtained.