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A right R-module N is called pseudo M-injective if for any submodule A of M, every monomorphism from A to N, can be extended to a homomorphism from M to N. Module M is called pseudo injective if M is pseudo M-injective. Some characterizations of classes modules and its applications to classical rings are studied. In this paper, we consider some generalizations pseudo injective modules under monomorphism of their closed submodules. Their properties are studied.

In this paper, the structure of e-local modules and classes of modules via essentially small are investigated. We show that the following conditions are equivalent for a module M :(1) M is e-local;(2) Rade(M ) is a maximal submodule of M and every proper essential submodule of M is contained in a maximal submodule;(3) M has a unique essential maximal submodule and every proper essential submodule of M is contained in a maximal submodule.

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