Weak Lifting Modules with Small Radical

Keskin and Tribak (2005) studied the structure of weak lifting modules with small radicals over commutative noetherian (local) rings. They proved that a module M is weak lifting if and only if M is a direct sum of local modules of some special type. Such modules were further studied by Tribak (2007). In this note we study weak lifting modules M with small radical over arbitrary rings. We prove that M is an irredundant sum M = Sigma(i is an element of I) M-i where each M-i is local and Sigma(i is an element of F) M-i is a summand of M for every finite subset F of I. Moreover Sigma(i is an element of F) M-i = circle plus(i is an element of F) K-i with K-i local. In particular a finitely generated weak lifting module is a direct sum of local modules. This generalizes the analogous result for finitely generated lifting modules.