左连续环中若干链条件的等价性

Equivalence of Some Conditions in Left Continuous Rings

（１）设Ｒ是左连续环，则Ｒ是在Ａｒｔｉｎ环当且仅当Ｒ满足左限制有限条件当且仅当Ｒ关于本质左理想满足极小条件当且仅当Ｒ关于本质左理想满足极大条件，同时给出一个左自内射环是ＱＦ环的充要条件；（２）证明了左 Ｚ１－环上的有限生成模都有 Ａｒｔｉｎ－Ｒｅｅｓ性质．

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-injec- tive ring is a QF ring. 2). It is proved that for a left Z1-ring R, if Mis a finitely generated R-module, then M satisfies Artin-Rees property.