GOOD IDEALS IN MATRIX RINGS OVER COMMUTATIVE PIR's

Abstract

An ideal A of a ring R is called a good ideal if the coset product r 1 r 2 + A of any two cosets r 1 + A and r 2 + A of A in the factor ring R/A equals their set product (r 1 + A) º (r 2 + A): = {(r 1 + a)(r 2 + a 2): a 1, a 2 ε A}. Good ideals were introduced in [3] to give a characterization of regular right duo rings. We characterize the good ideals of blocked triangular matrix rings over commutative principal ideal rings and show that the condition A º A = A is sufficient for A to be a good ideal in this class of matrix rings, none of which are right duo. It is not known whether good ideals in a base ring carries over to good ideals in complete matrix rings over the base ring. Our characterization shows that this phenomenon occurs indeed for complete matrix rings of certain sizes if the base ring is a blocked triangular matrix ring over a commutative principal ideal ring.