Generalizations of Multiplicative Ideal Theory to Commutative Rings with Zerodivisors

Abstract

Multiplicative ideal theory has at first been developed for (commutative) integral domains. We concern here generalizations of the theory to (commutative) rings with zerodivisors. At first, Manis [50] defined a valuation for a commutative ring with zerodivisors, and generalized basic properties of a valuation on a domain(1). Using the results of Manis, Griffin [35] extended the notion of prufer domain for commutative rings with zerodivisors, and extended conditions under which a ring is a prufer ring. And Larsen generalized the notion and properties of almost Dedekind domain for rings with zerodivisors [46], generalized primary ideal structure of a prufer domain for a ring with zerodivisors. Also he extended the notion of finite character and characterizations of a prufer domain with finite character for a ring with zerodivisors [47]. Next Hinkle-Huckaba [38] defined a Kronecker function ring for a ring with zerodivisors and generalized a property of a Kronecker function domain(2). Besides, Kennedy [43] extended the notion of Krull domain for a ring with zerodivisors and generalized some properties of a Krull domain for a ring with zerodivisors(3). Here we generalize all of multiplicative ideal theory for a ring with zerodivisors. The subjects remaining for generalizations are as follows: 1. We know by Griffin [32] the extension of conditions of a prufer domain to a prufer *-multiplication domain, and a relationship between a prufer v-multiplication domain and a domain of Krull type.