ENDOMORPHISM RINGS OF QUASI-PRINCIPALLY INJECTIVE MODULES

Abstract

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.

Keywords:

• Quasi p-injective module
• Self generator
• Annihilator
• 1991 Mathematics Subject Classification: 16D50, 16D70, 16D80

ACKNOWLEDGMENT

The authors would like to thank Professor R. Wisbauer and the referee for many helpful comments. Nguyen Van Sanh gratefully acknowledges the support of the Department of Mathematics, The Chinese University of Hong Kong, for his visit to Hong Kong. He also would like to thank The Thailand Research Fund for a partial support during the preparation of this paper. This research is partially supported by a CUHK small project grant # 2060152.