On Left Quasi Noetherian Rings

Abstract: In this paper we prove; If R is a left quasi-Noetherian ring,then every nil subring is nilpotent). Next we show that a commutative semi-prime quasi-Noetherian ring is Noetherian. Then we study the relationship between left Quasi-Noetherian and left Quasi-Artinian, in particular we prove that If R is a non-nilpotent left Quasi-Artinian ring. Then any left R-module is left Quasi-Artinian if and only if it is left Quasi-Noetherian. Finally we show that a commutative ring R is Quasi-Artinian if and only ifR is Quasi-Noetherian and every proper prime ideal of R is maximal