SP-INJECTIVITY OF MODULES AND RINGS

Let >M and N be two R-modules. NR is called singular M-p-injective if for every singular M-cyclic submodule X of MR, every homomorphism from X to N can be extended to a homomorphism from M to N. MR 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-sp-injective modules in terms of their endomorphism rings.