Quasipolar Subrings of 3 x 3 Matrix Rings

An element a of a ring R is called quasipolar provided that there exists an idempotent p ∈ R such that p ∈ comm²(a), a + p ∈ U (R) and ap ∈ B^qnil. A ring R is quasipolar in case every element in R is quasipolar. In this paper, we determine conditions under which subrings of 3 x 3 matrix rings over local rings are quasipolar. Namely, if R. is a bleached local ring, then we prove that T₃ (R) is quasipolar if and only if R is uniquely bleached. Furthermore, it is shown that T_n(R) is quasipolar if and only if T_n(R[[x]]) is quasipolar for any positive integer