Kronecker Function Rings

Abstract

0. Introduction. Let D be an integral domain, K the quotient field of D, and K(X) the rational function field of one variable X over K. The Kronecker function rings D* which are interesting subrings of K(X) were first defined by Prufer in [25] and further studied by Krull in [16]. After that the notion of Kronecker function rings has been used as a tool to study finite characters (cf. Brewer-Mott [3]), contraction of ideals (cf. Gilmer-Mott [9]) and the stable range (cf. Estes-Ohm [4]). We have also results on D* given by Arnold, Brewer, Gilmer and Mott. For commutative ring A with zerodivisors, Hinkle-Huckaba [12] defined a special Kronecker function ring Ab and generalized a part of the results of Arnold [1]. In this paper we define general Kronecker function rings A* and show that fundamental properties of D* hold for our A*. And then we prove analogies of the results of Arnold, Brewer, Gilmer and Mott to rings with zerodivisors. This paper consists of 5 sections. In 1, we study fundamental properties of the Kronecker function rings for rings with zerodivisors. In 2, we study analogies of the results of Arnold, Gilmer-Mott on the contractions and the extensions of ideals. In 3, we prove an analogy of Theorem 5 of Arnold [1] on almost Dedekind rings. In 4, we prove an analogy of Theorem 6 of Arnold [1] on Dedekind rings. In final 5, we study conditions under which A is a prufer *-multiplication ring. Let R be a commutative ring (not necessarily a domain). We call a nonzerodivisor of a ring a regular element, and we call an ideal containing a regular element