Some results on self-injective rings and Σ-CS rings

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 R-R((N)) is CS, then R-R((A)) is also CS for any set A. Moreover, R is then right and left artinian.