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In this note, we introduce the (small, pseudo-)ℬ(M,X)-cojective modules and we generalize (small, pseudo-)cojective modules via the class ℬ(M,X). Let M = M1 ⊕ M2 be an X-amply supplemented module with the finite internal exchange property. Then for every decomposition of M = Mi ⊕ Mj, Mi is ℬ(Mj,X)-cojective for i ≠ j, M1 and M2 are X-lifting if and only if M is X-lifting. We also prove that for an X-amply supplemented module M = M1 ⊕ M2 such that M1 and M2 are indecomposable X-lifting modules, if M2 is ℬ(M1,X)-cojective and M1 is small-ℬ(M2,X)-cojective then M is X-lifting.

A module M is called ⊕-supplemented if every submodule of M has a supplement that is a direct summand of M. It is shown that if M is a ⊕-supplemented module and r(M) is a coclosed submodule of M for a left preradical r, then r(M) is a direct summand of M, and both r(M) and M/r(M) are ⊕-supplemented.

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