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Modules which are invariant under idempotents of their envelopes

Le Van ThuyetPhan DanTruong Cong Quynh — 2016

Colloquium Mathematicae

We study the class of modules which are invariant under idempotents of their envelopes. We say that a module M is -idempotent-invariant if there exists an -envelope u : M → X such that for any idempotent g ∈ End(X) there exists an endomorphism f : M → M such that uf = gu. The properties of this class of modules are discussed. We prove that M is -idempotent-invariant if and only if for every decomposition X = i I X i , we have M = i I ( u - 1 ( X i ) M ) . Moreover, some generalizations of -idempotent-invariant modules are considered....

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