Equivalence of Some Conditions in Left Continuous Rings

Abstract:

      In this paper, we obtain following results: 1). Let R be a left continuous ring, then R be a left Artinian iff R satisfies left restricted finite condition iff R satisfies DCC on essential left ideals iff R satisfies ACC on essential left ideals. In addition we give a sufficient and necessary condition under which a left self-injective ring is a QF ring.2). It is proved that for a left Z₁-ring R, if M is a finitely generated R-module, then M satisfies Artin-Raes property.