A Result on Semi-Artinian Rings

Abstract

It was shown by Huynh and Rizvi that a ring R is semisimple artinian if and only if every continuous right R-module is injective. However, a characterization of rings, over which every finitely generated continuous right module is injective, has been left open. In this note we give a partial solution for this question. Namely, we show that for a right semi-artinian ring R, every finitely generated continuous right R-module is injective if and only if all simple right R-modules are injective. (p) Every finitely generated continuous right R-module is injective.(q) Every finitely generated CS right R-module is (quasi-) continuous.The structure of rings satisfying either (p) or (q) is unknown. However, such a ring need not be semisimple artinian. It is shown in [9] that any simple right and left SI ring with 63