Abstract
Direct complements in a right R-module M are said to be almost unique if, whenever M = A circle plus B = A circle plus C, then (B + C)/ B << M/B (also (B + C)/C << M/C, by symmetry). We will show that this new class of modules lies strictly between the dual-square-free and the summand-dual-square-free modules. While it is an open question whether right quasi-duo (equivalently, right dual-square-free) rings are left-right symmetric, we will prove that both notions "summand-dual-square-free" and "direct complements almost unique" are left-right symmetric. Furthermore, we will show that direct complements in a module M are almost unique iff idempotents in S := End(MR) are central modulo del(M), where del(M) := {f is an element of S : Im f << M}. As an immediate consequence, if M is an epi-projective module, then direct complements in M are almost unique iff M is strongly perspective (i.e. if A and B are isomorphic direct summands of M and M = A circle plus X, then M = B circle plus X). in particular, direct complements in a ring R are almost unique iff R is strongly perspective. Moreover, if M is a module with the finite exchange whose direct complements are almost unique, then M is clean, strongly perspective, and satisfy the full exchange.