Some Results on Self-Injective Rings and Σ-CS Rings

Abstract

A module M is CS if every submodule of M is essential in a direct summand of M. In this note we use the CS condition to provide conditions for semiperfect rings to be self-injective. Further we show that every finitely generated CS right module over a right semi-artinian ring has finite uniform dimension. Using this, we prove that if R is a right semi-artinian ring such that is CS, then is also CS for any set A. Moreover, R is then right and left artinian.

Key Words:

• CS
• (Quasi)-continuous
• Injective modules
• Self-injective
• Quasi-Frobenius rings

(2000) Mathematics Subject Classification:

• 16D50
• 16L60
• 16P20

Acknowledgments

The authors wish to express their gratitude to the referee for many useful comments and suggestions. Especially they are grateful to the referee for pointing out an important part that was missing in the proof of Lemma 1.1.

Hai Quang Dinh was partially supported by Center of Ring Theory and Applications, Ohio University.