STRONGLY CLEAN TRIANGULAR MATRIX RINGS WITH ENDOMORPHISMS

A ring R is strongly clean provided that every element in R is the sum of an idempotent and a unit that commutate. Let T-n(R, sigma) be the skew triangular matrix ring over a local ring R where a is an endomorphism of R. We show that T-2(R, sigma) is strongly clean if and only if for any a is an element of 1 + J(R),b is an element of J(R), l(a) - r(sigma(b)) : R -> R is surjective. Further, T-3(R, sigma) is strongly clean if l(a) - r(sigma(b)), l(a) - r(sigma 2(b)) and l(b) - r(sigma(a)) are surjective for any a is an element of U(R), b is an element of J(R). The necessary condition for T-3(R, sigma) to be strongly clean is also obtained.