Abstract
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.
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