PSEUDO QP-INJECTIVE MODULES AND GENERALIZED PSEUDO QP-INJECTIVE MODULES

Let M be a right R -module with S = End(M-R). Then MR is called pseudo QP-injective (or PQP-injective for short) if every monomorphism from an M -cyclic submodule of M to M extends to an endomorphism of M. M-R is called generalized pseudo QP-injective (or GPQP-injective for short) if, for any 0 not equal s is an element of S, there exists a positive integer n such that s(n) not equal 0 and every monomorphism from s(n) M to M extends to an endomorphism of M. Characterizations and properties of the two classes of modules are studied. The two classes of modules with some additional conditions are studied, semisimple artinian rings are characterized by PQP-injective modules.