Seminar:  OSU-OU RING THEORY SEMINAR

Title:    A generalization of Kronecker function rings and Nagata rings

Speaker:  Alan Loper, The Ohio State University -Newark

Day/Date: Friday, April 13, 2007

Time:     4:50 PM

Room:  MW154

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

special cases of the one general construction.

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.

This is joint work with Marco Fontana.

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