Nagata Rings, Kronecker Function Rings, and Related Semistar Operations

Abstract

In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer's book (Gilmer, R. (1972). Multiplicative Ideal Theory. New York: Marcel Dekker) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer and P. Lorenzen from 1930's. Fontana and Loper investigated properties of the Kronecker function rings which arise from arbitrary semistar operations on an integral domain D (Fontana M., Loper K. A. (2001a). Kronecker function rings: a general approach. In Anderson, D. D., Papick, I. J., eds. Ideal Theoretic Methods in Commutative Algebra. Lecture Notes Pure Appl. Math. 220, Marcel Dekker, pp. 189–205 and Fontana, M., Loper, K. A. (2001b). A Krull-type theorem for the semistar integral closure of an integral domain. ASJE Theme Issue “Commutative Algebra” 26:89–95). In this paper we extend that study and also generalize Kang's notion of a star Nagata ring to the semistar setting (Kang, B. G. (1987). ⋆-Operations on Integral Domains. Ph.D. dissertation, Univ. Iowa and Kang, B. G. (1989). Prüfer v-multiplication domains and the ring R[X] _{N v} , J. Algebra 123: 151–171). Our principal focuses are the similarities between the ideal structure of the Nagata and Kronecker semistar rings and between the natural semistar operations that these two types of function rings give rise to on D.

Acknowledgments

During the preparation of this work the first named author was partially supported by a research grant MIUR 2001/2002 (Cofin 2000 - MM 01192794). Fontana thanks also the Mathematical Department of the Ohio State University for the hospitality accorded during his visit in July 2001.