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On small injective, simple-injective and quasi-Frobenius rings.

Thuyet, Le Van; Quynh, Truong Cong — 2009

Acta Mathematica Universitatis Comenianae. New Series

Modules which are invariant under idempotents of their envelopes

Le Van Thuyet; Phan Dan; Truong 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 $X = ⨁_{i \in I} X_{i}$ , we have $M = ⨁_{i \in I} (u^{- 1} (X_{i}) \cap M)$ . Moreover, some generalizations of -idempotent-invariant modules are considered....

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