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AbstractIn this paper, we study modules having the property that are invariant under some idempotent endomorphisms of its injective envelope. Such modules are called essentially π-injective. It is shown that (1) M is essentially π-injective iff for any essentially finite direct summand X1 of M and any submodule X2 of M with X1∩X2=0 , there exists a direct summand X0 of M containing X2 such that M=X1⊕X0 , (2) M is essentially π-injective iff M is an ef-extending right R-module and for any decomposition M=M1⊕M2 with M1 essentially finite, M1 and M2 are relatively injective, (3) if M is essentially π-injective and R satisfies ACC on right ideals of the form r(m), m∈M , then M is a direct sum of uniform submodules. We also describe rings via essentially π-injective modules. It is shown that R is a semisimple artinian ring iff the direct sum of any two essentially π-injective right R-modules is essentially π-injective. [ABSTRACT FROM AUTHOR]

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]

A right R-module M is called semi-projective if, for any submodule N of M, every epimorphism π : M → N and every homomorphism α : M → N, there exists a homomorphism ß : M → M such that πß = a (see [11]). In this paper, we consider some generalizations of semi-projective module, that is quasi pseudo principally projective module. Some properties of this class of module are studied. [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]

Let R be a ring. A right ideal I of R is called small in R if I + K ≠ R for every proper right ideal K of R. A ring R is called right small finitely injective (briefly, SF-injective) (resp., right small principally injective (briefly, SP-injective) if every homomorphism from a small and finitely generated right ideal (resp., a small and principally right ideal) to RR can be extended to an endomorphism of RR. The class of right SF-injective and SP-injective rings are broader than that of right small injective rings (in [15]). Properties of right SF-injective rings and SP-injective rings are studied and we give some characterizations of a QF-ring via right SF-injectivity with ACC on right annihilators. Furthermore, we answer a question of Chen and Ding. [ABSTRACT FROM AUTHOR]

Kupisch proved that if R is a left and right artinian QF-2 ring and Sr = Sl, then R is QF. A weaker condition for a ring to be a QF ring was obtained by Dan and Thuyet. They proved that if R is a right artinian QF-2 ring and Sr ≤ Sl, then R is QF. In this paper, we prove that if R is a QF-2 ring satisfying ACC on right annihilators in which Sl ≤ eRR (e.g., Sr ≤ Sl with Sr ≤e RR), then R is QF. It is also proved that R is QF if and only if R is a left ef-extending, right continuous ring with ACC on right annihilators. [ABSTRACT FROM AUTHOR]

A right R-module MR is called small injective if every homomorphism from a small right ideal to MR can be extended to an R-homomorphism from RR to MR. A ring R is called right small injective, if the right R-module RR is small injective. We prove that R is semiprimitive if and only if every simple right (or left) R-module is small injective. Further we show that the Jacobson radical J of a ring R is a noetherian right R-module if and only if, for every small injective module ER, E(ℕ) is small injective. [ABSTRACT FROM AUTHOR]