Separative Ideals of Exchange Rings

An ideal I of an exchange ring R is separative provided that for all A,B 2 FP(I), 2A cong A oplus B cong 2B implies that A cong B. We prove that I is separative if and only if so is the ideal of all (triangular) matrices over I. Further, we investigate diagonal reduction over such ideals. Comparability of modules over such ideals are studied as well.