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 The strong Mori property in rings with zero divisors Bull. Korean Math. Soc. 2015 Vol. 52, No. 4, 1285-1295 https://doi.org/10.4134/BKMS.2015.52.4.1285Published online July 31, 2015 Dechuan Zhou and Fanggui Wang Sichuan Normal University, Sichuan Normal University 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. Keywords : $Q_{0}$-SM ring, semiregular $w$-ideal, semiregular $v$-ideal MSC numbers : 13A15, 13F05 Downloads: Full-text PDF