Sp-Injectivity of Modules and Rings

Abstract

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...