Direct Complements Almost Unique

Abstract

Direct complements in a right [Formula: see text]-module [Formula: see text] are said to be almost unique if, whenever [Formula: see text], then [Formula: see text] (also [Formula: see text], 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 [Formula: see text] are almost unique iff idempotents in [Formula: see text] are central modulo [Formula: see text], where [Formula: see text]. As an immediate consequence, if [Formula: see text] is an epi-projective module, then direct complements in [Formula: see text] are almost unique iff [Formula: see text] is strongly perspective (i.e. if [Formula: see text] and [Formula: see text] are isomorphic direct summands of [Formula: see text] and [Formula: see text], then [Formula: see text]. In particular, direct complements in a ring [Formula: see text] are almost unique iff [Formula: see text] is strongly perspective. Moreover, if [Formula: see text] is a module with the finite exchange whose direct complements are almost unique, then [Formula: see text] is clean, strongly perspective, and satisfy the full exchange.