T-idempotent invariant modules

Abstract

We introduce and investigate t-idempotent invariant modules. We call an endomorphism ψ of M, a t-idempotent endomorphism if ψδ: M/Z2(M) → M/Z2(M) defined by ψδ(m + Z2(M)) = ψ(m) + Z2(M) is an idempotent and we call a module M is t-idempotent invariant, if it is invariant under t-idempotents of its injective envelope. We prove a module M is t-idempotent invariant if and only if M = Z2(M) M′, Z2(M) is quasi-injective, M′ is quasi-continuous and Z2(M) is M′-injective. The class of rings R for which every (finitely generated, cyclic, free) R-module is t-idempotent invariant is characterized. Moreover, it is proved that if R is right q.f.d., then every t-idempotent invariant R-module is quasi-injective exactly when every nonsingular uniform R-module is quasi-injective.