Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: $Q_{0}$-SM ring, semiregular $w$-ideal, semiregular $v$-ideal

MSC numbers: 13A15, 13F05