On ?-Morphıc Modules

Abstract

Let R be an arbitrary ring with identity and M be a right R-module with S = End(MR). Let f ? S. f is called ?-morphic if M/f n (M) ?=rM(fn) for some positive integer n. A module M is called ?-morphic if every f ? S is ?-morphic. It is proved that M is ?-morphic and image-projective if and only if S is right ?-morphic and M generates its kernel. S is unit-?-regular if and only if M is ?-morphic and ?-Rickart if and only if M is ?-morphic and dual ?-Rickart. M is ?-morphic and image-injective if and only if S is left ?-morphic and M cogenerates its cokernel.

Let R be an arbitrary ring with identity and M be a right R-module with S = End(MR). Let f ? S. f is called ?-morphic if M/f n (M) ?=rM(fn) for some positive integer n. A module M is called ?-morphic if every f ? S is ?-morphic. It is proved that M is ?-morphic and image-projective if and only if S is right ?-morphic and M generates its kernel. S is unit-?-regular if and only if M is ?-morphic and ?-Rickart if and only if M is ?-morphic and dual ?-Rickart. M is ?-morphic and image-injective if and only if S is left ?-morphic and M cogenerates its cokernel.