T-idempotent invariant modules

We introduce and investigate [Formula: see text]-idempotent invariant modules. We call an endomorphism [Formula: see text] of [Formula: see text], a [Formula: see text]-idempotent endomorphism if [Formula: see text] defined by [Formula: see text] is an idempotent and we call a module [Formula: see text] is [Formula: see text]-idempotent invariant, if it is invariant under [Formula: see text]-idempotents of its injective envelope. We prove a module [Formula: see text] is [Formula: see text]-idempotent invariant if and only if [Formula: see text], [Formula: see text] is quasi-injective, [Formula: see text] is quasi-continuous and [Formula: see text] is [Formula: see text]-injective. The class of rings [Formula: see text] for which every (finitely generated, cyclic, free) [Formula: see text]-module is [Formula: see text]-idempotent invariant is characterized. Moreover, it is proved that if [Formula: see text] is right q.f.d., then every [Formula: see text]-idempotent invariant [Formula: see text]-module is quasi-injective exactly when every nonsingular uniform [Formula: see text]-module is quasi-injective.

Primary Groups Whose Basic Subgroup Decompositions can be Lifted

A primary group G is said to be a l.i.b. group if every idempotent endomorphism of every basic subgroup of G can be extended to an endomorphism of G. We establish the following characterization: A primary group is a l.i.b. group if and only if it is the direct sum of a torsion complete group and a divisible group. The technique used consists of a close analysis of certain subgroups of Prufer-like primary groups.