The Strong Mori Property In Rings With Zero Divisors

Abstract:An SM domain(a strong Mori domain) is an important general-ization of the Noetherian domain and it has many nice properties as same as the Noetherian domain’s. Naturally, we want to know the properties of SM domains responding to commutative rings. In this paper, we use semiregular ideals to study commutative rings. And a Q0-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. Let R be a commutative ring and let P be a prime ideal of R, by discussing the reducible property of R and its Q0-quotient ring R〈P〉 with respect to P, we get the equivalent depiction of Q0-SM rings similar to SM domains which is that if R is a Q0-SM ring if and only if each semiregular ideal of R is contained in at most finitely many semiregular maximal w-ideals and R〈m〉 is a Q0m-Noetherian ring for each semiregular maximal w-ideal m. Then by defining and discussing the w-global transformation ring Rw*of a commutative ring R, we obtain a similar Matijevic Theorem:If R is a Q0-SM ring, so is Rw*and t-dim(Rw*)=t-dim(R)-1. At last, we define semireg-injective modules, the semireg-injective hull e0(M) of an R-module M and∑-semireg-injective modules, and then we give the generalization of Fuchs’s result:If each semiregular v-ideal of R contains a v-invertible ideal, then R is a Q0-SM ring if and only ie c0(Q0(R)/R) is∑-semireg-injective.