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A submodule N of M is called fully invariant if f(N) ≤ N for all f ∈ End(M). In this paper, we consider a generalization of fully invariant submodules and its applications. In particular, regularity of endomorphism ring of finitely generated quasi projective modules are studied. Let M be a quasi-projective module and S = End(M). We show that S is regular if and only if every right R-module is M-gp-injective. [ABSTRACT FROM AUTHOR]

Let R be a ring. Let m and n be positive integers, a right R-module M is called (m,n)-small injective, if every right R-homomorphism from an n-generated submodule of Jm to M extends to one from Rm to M. A ring R is called right (m,n)-small injective if the right R module RR is (m,n)-small injective. In this paper, we give some properties of (m,n)-small injective modules and right (m,n)-small injective rings. [ABSTRACT FROM AUTHOR]