CERTAIN ADDITIVE DECOMPOSITIONS IN A NONCOMMUTATIVE RING

We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a 2 x 2 matrix A over a projective-free ring R is strongly J-clean if and only if A is an element of J(M-2(R)), or I-2 - A is an element of J(M-2(R)), or A is similar to [GRAPHICS] , where lambda is an element of J(R), mu is an element of 1 + J(R), and the equation x(2) - x mu - lambda = 0 has a root in J(R) and a root in 1 + J(R). We further prove that f(x) is an element of R[[x]] is strongly J-clean if f(0) is an element of R be optimally J-clean.