When Socles Split in Injectivity Domains of Modules

In this paper, we investigate modules whose elements of their injectivity domains satisfy the fixed property p. In particular, quasi poor modules as a generalization of poor modules are introduced and investigated as modules that every element of their injectivity domains is a socle-split module. We evaluate the feasibility that all modules are either quasi-poor or injective and characterize the rings satisfying this property (this property will be referred to as property (*)). Some descriptions of known rings in terms of quasi-poor modules are given. We completely determine right non-singular rings that have the property (*) and we show that a right non-singular ring R satisfies the property (*) iff R = R(1 )x R-2 where R-1 is semisimple Artinian and either R-2 is zero or soc(R-2) = 0 and R-2 is a right (SI)-I-3-ring or R-2 is a right (SI)-I-3-ring with essential homogeneous right socle and any proper essential submodule of its maximal right quotient ring is quasi-poor. Also, it is shown that if R satisfies the property (*) and Z(2)(R) &NOTEQUexpressionL;0, then R/Z(2)(R) is right Noetherian and a right V-ring, by using hereditary pre-torsion classes in the category of right R-modules. Further, it is proved that a ring R with Z(2)(R)&NOTEQUexpressionL; 0 satisfies the property (*) iff R = R-1 x R-2, where R-1 is semisimple and Artinian and R2 = Z(2)(R) satisfies the property (*) and soc(R-2) is singular.