OUOSU ring theory seminar, Alan Loper, April 13 2007

Abstract: Let $D$ be an integrally closed integral domain. The Kronecker function ring $Kr(D)$ of $D$ and the Nagata ring $D(X)$ of $D$ are both overrings of the ring $D[X]$ of polynomials over $D$ with numerous nice properties. The Nagata ring is always contained in the Kronecker function ring, but the two are usually distinct. Both of these domains have been generalized by many people but the two types of domains and their generalizations are treated in the literature as two very distinct objects of study. In this talk we give a single construction which generalizes the construction of the Nagata ring and the Kronecker function ring. In particular, $Kr(D)$ and $D(X)$ become

Kronecker's original work dealt with the setting where the domain $D$ in question was a ring of algebraic integers. His work was generalized tremendously by Krull to any integrally closed domain by means of star operations satisfying a property he called a.b. (This property was later weakened by Gilmer to a finite version labelled the e.a.b. property.) One of the major tools of our generalization is to consider the e.a.b. property as a property of a class of ideals rather than as a property of star operations. In this view the classical e.a.b. star operations seem especially attractive becasuse relative to them all finitely generated ideals are e.a.b., which makes possible the Kronecer function ring construction. On the other hand, only invertible ideals are e.a.b. ideals for the identity operation. In this very restrictive setting the generalized construction yields the classical Nagata ring. So we can explore star operations which are associated with classes of e.a.b. ideals which consist of more than just the invertible ideals and yet not all of the finitely generated ideals. The result is a class of rings which lie in between the Nagata and Kronecker rings and yet have many properties in common with these two rings.

With the same {\it opposite ends of a spectrum} reasoning we consider that both the Nagata and Kronecker function ring can be defined using particular quailocal overrings. In the case of the Nagata ring, these overrings are localizations of the domain $D$ and in the case of the Kronecker function ring, these overrings are valuation domains. So we explore classes of domains which lie in between the class of localizations and the class of valuation overrings to see which intermediate classes correspond to our generalized function rings.