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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 $M={⨁}_{i\in I}\left(u^{-1}\left(X_i\right)\cap M\right)$. Moreover, some generalizations of -idempotent-invariant modules are considered....