The Strong Mori Property in Rings With Zero Divisors

Abstract

Abstract. An SM domain is an integral domain which satisfies the ascending chain condition on w-ideals. Then an SM domain also satisfies the descending chain condition on those chains of v-ideals whose intersection is not zero. In this paper, a study is begun to extend these properties to commutative rings with zero divisors. A Q 0 -SM ring is defined to be a ring which satisfies the ascending chain condition on semiregular w-ideals and satisfies the descending chain condition on those chains of semiregular v-ideals whose intersection is semiregular. In this paper, some properties of Q 0 -SM rings are discussed and examples are provided to show the difference between Q 0 -SM rings and SM rings and the difference between Q 0 -SM rings and Q 0 -Mori rings.