Left Quasi- ArtinianModules

In this paper we study a new class of left quasi-Artinian modules. we show: if R is a left quasi-Artinian ring and M is a left R-module, then (a) Soc(M) ess M and (b) Rad(M) small in M .Then we prove: if I is a non-nilpotent left ideal in a left quasi-Artinian ring, then I contains a non-zero idempotent element. Finally we show that a commutative ring R is quasi-Artinian if and only if R is a direct sum of an Artinian ring with identity and a nilpotent ring.