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 <TEX>$Q_0$</TEX>-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 <TEX>$Q_0$</TEX>-SM rings are discussed and examples are provided to show the difference between <TEX>$Q_0$</TEX>-SM rings and SM rings and the difference between <TEX>$Q_0$</TEX>-SM rings and <TEX>$Q_0$</TEX>-Mori rings.
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