ENDOMORPHISM RINGS OF QUASI-PRINCIPALLY INJECTIVE MODULES

A right R-module M is called quasi-principally injective if every homomorphism from an M-cyclic submodule of M to M can be extended to M. Let M be a quasi-principally injective module which a self generator. In this paper we show that if such a module M has finite Goldie dimension, then S/J(S) is semisimple, where S = End(M R ). Furthermore if M is a self-generator quasi-principally injective module and M/soc(M) satisfies ACC on M-annihilator submodules, then J(S) is nilpotent. Some recent results obtained by Nicholson and Yousif are generalized.