In this paper we study M -small principally injective (in short, M -sp-injective) module which is the generalization of M -principally injective module. We prove that if M is finite dimensional and quasi-sp-injective then its endomorphism ring S is semi-local ring. We characterize the M -sp-injective module with the help of epi-retractable modules. Keywords: Small submodule; M -cyclic submodule; M -sp-injective modules; quasi-spinjective modules and epi-retractable module. AMS Subject Classification: 16D10, 16D50, 16D60 1250005-1 V. Kumar et al.N . A module M is called quasi-principally (or semi) injective, if it is M -principally injective. In this paper we introduce the notion of M -small principally injective modules and quasi-small principally injective modules which we abbreviate as Msp-injective and quasi-sp-injective modules. A submodule K of an R-module M is said to be small in M , written as K M , if for every submodule L ⊆ M withFor details see [1,5]. In this paper, we study M -sp-injective modules and M -sp-injective rings and also give an example of an M -sp-injective module which is not an M -principally injective module.Consider the following condition for an R-module M ., it is called continuous, if it satisfies (C 1 ) and (C 2 ), and quasi-continuous, if it satisfies (C 1 ) and (C 3 ). For undefined notations and terminologies see [1,14]. M-Small Principally Injective ModulesGeneralizing the notion of Sanh et al. [13], we now give a new definition. DefinitionAn R-module N is called M -small principally injective, if for every small M -cyclic submodule K of M , any homomorphism from K to N can be extended to a homomorphism from M to N . If M is a M -small principally injective module, then it is called a quasi-small principally injective module, or M -sp-injective module, and for short a ring R is called a right small-principally injective ring, if R R as a right R-module. Example(1) Every M -principally injective module is a M -sp-injective module.(2) Every semi-simple module is an M -sp-injective module.We now give an example of M -sp-injective modules which is not M -principally injective. ExampleZ is a Z-small principally injective module, but it is not Z-principally injective, because the only small Z-cyclic submodule of Z is 0.The following lemmas are the generalizations of [13, Lemmas 2.2-2.4]. Lemma 2.1. Let M i (1 ≤ i ≤ n) be M-sp-injective modules. Then n i=1 M i is M -sp-injective module. 1250005-2 M -SP-Injective Modules Proof. The proof is similar to that of [13, Lemma 2.2]. Lemma 2.2. Let X be an M-cyclic submodule of M . If X is M-sp-injective module, then it is a direct summand of M . Proof. The proof is similar to that of [13, Lemma 2.3]. Lemma 2.3. Every direct summand of M-sp-injective module is an M-sp-injective module. Proof. By the same argument as that given in [13, Lemma 2.4]. 1250005-5 V. Kumar et al. Proposition 2.6. For a hollow R-module M, the following conditions are equivalent :(1) M is quasi-sp-injective module;(2) M is quasi-principally injective module.Proof. It is ...

In this paper, the concept of quasi-pseudo principally injective modules is introduced and a characterization of commutative semi-simple rings is given in terms of quasi-pseudo principally injective modules. An example of pseudo M -p-injective module which is not M -pseudo injective is given.

Let M1 and M2 be two R-modules. Then M2 is called M1-c-injective if every homomorphism α from K to M2, where K is a closed submodule of M1, can be extended to a homomorphism β from M1 to M2. An R-module M is called self-c-injective if M is M-c-injective. For a projective module M, it has been proved that the factor module of an M -c-injective module is M -c-injective if and only if every closed submodule of M is projective. A characterization of self-c-injective modules in terms endomorphism ring of an R-module satisfying the CM-property is given.

In this paper characterization of pseudo M -p-projective modules and quasi pseudo principally projective modules are given and discussed the various properties of it. It is proved that a pseudo M -pprojective module is Hopfian iff M/N is Hopfian, for each fully invariant small submodule N of M . It is also provided the sufficient condition for pseudo M -p-projective module to be discrete.Finally several equivalent conditions are given for a quasi pseudo principally projective module to have the finite exchange property.

Let M and N be two R-modules. N R is called singular M-p-injective if for every singular M-cyclic submodule X of M R , every homomorphism from X to N can be extended to a homomorphism from M to N . M R is quasi-singular prinicipally injective if M is a singular M-p-injective module. It is shown that a ring R is right non-singular if and only if every right R-module is singular R-p-injective if and only if factors of singular R-p-injective modules are singular R-p-injective. A singular R-module M is injective if and only if M is N-sp-injective for every R-module N . Finally, we characterize quasi-spinjective modules in terms of their endomorphism rings. Asian-European J. Math. 2012.05. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/27/15. For personal use only. 1250053-2 Asian-European J. Math. 2012.05. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/27/15. For personal use only. SP-Injectivity of Modules and Rings Lemma 2.2. Let A, B and X be R-modules with A isomorphic to B. If A is singular X-p-injective, then B is singular X-p-injective. Lemma 2.3. Let X, Y and M be R-modules withProof. The proof is similar to [9, Proposition 2.2].Proof. Let X be a singular B-cyclic submodule of B. Then X is a singular Acyclic submodule of A. Since M is singular A-p-injective, therefore obviously M is singular B-p-injective. Let f : X → N be a homomorphism, j 1 : N → M be the inclusion and π 1 : M → N be canonical projection. Suppose i : X → B is the inclusion map. Since M is singular B-p-injective, therefore there exists a homomorphism g : B → M such that j 1 f = gi. It follows that π 1 j 1 f = π 1 gi. This implies that If = hi, where I = π 1 j 1 and h = π 1 g : B → N . Hence N is singular B-p-injective. Corollary 2.7. Direct summand of every quasi-sp-injective module is quasi-spinjective. Corollary 2.8. If M is a quasi-sp-injective module and X is a singular M-cyclic submodule of M, then M is singular X-p-injective.All modules X j , j ∈ A are called relatively sp-injective modules if X j is singular X k -p-injective for all distinct j, k ∈ A, where A is some index set. Corollary 2.9. If i∈A X i is quasi-sp-injective, where A is finite index set, then X j and X k are relatively sp-injective for all distinct j, k ∈ A. Corollary 2.10. If M n is quasi-sp-injective for any finite integer n, then M is quasi-sp-injective.Proof. It follows from Proposition 2.4 and Proposition 2.6.We recall (C 2 ) condition: If a submodule of an R-module M is isomorphic to a direct summand of M , then it is itself a direct summand of M . Also, (C 3 ) condition: 1250053-3 Asian-European J. Math. 2012.05. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/27/15. For personal use only. A. J. Gupta, B. M. Pandeya & A. K. Chaturvedi 1250053-4 Asian-European J. Math. 2012.05. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/27/15. For personal use only. SP-Injectivity of Modules and RingsNow, we investigate when a singular mo...