K. Alan Loper

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, Gilmers 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 1930s. 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 Kangs 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.

An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations

An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations

The content of a Gaussian polynomial is invertible

Let R be an integral domain and let f(X) be a nonzero polynomial in R[X]. The content of f is the ideal c(f) generated by the coefficients of f. The polynomial f(X) is called Gaussian if c(fg) = c(f)c(g) for all g(X) ∈ R[X]. It is well known that if c(f) is an invertible ideal, then f is Gaussian. In this note we prove the converse.

The Patch Topology and the Ultrafilter Topology on the Prime Spectrum of a Commutative Ring

Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well-known patch or constructible topology. The proof is accomplished by use of a von Neumann regular ring canonically associated with R.

A GENERALIZATION OF KRONECKER FUNCTION RINGS AND NAGATA RINGS

Abstract Let D be an integral domain with quotient field K. The Nagata ring D(X) and the Kronecker function ring Kr(D) are both subrings of the field of rational functions K(X) containing as a subring the ring D[X] of polynomials in the variable X. Both of these function rings have been extensively studied and generalized. The principal interest in these two extensions of D lies in the reflection of various algebraic and spectral properties of D and Spec(D) in algebraic and spectral properties of the function rings. Despite the obvious similarities in definitions and properties, these two kinds of domains of rational functions have been classically treated independently, when D is not a Prüfer domain. The purpose of this note is to study two different unified approaches to the Nagata rings and the Kronecker function rings, which yield these rings and their classical generalizations as special cases.

On Prüfer non-D-rings

Abstract Call a commutative integral domain R with unity a non-D-ring if there exists a nonconstant polynomial f ( x ) over R such that f ( a ) is a unit in R for all a in R . This paper is concerned with Prufer non-D-rings which have a monic unit valued polynomial. First, we prove that the ideal class group of such a ring is torsion. Secondly, we show that such rings can be precisely characterized as intersections of non-D valuations domains which share a common monic unit valued polynomial. Finally, let R be a Prufer non-D-ring with a monic unit valued polynomial and with field of quotients K . Let W ( R ) be the set of all rational functions f ( x ) / g ( x ) in K ( x ) such that f ( a ) / g ( a ) is in R for all a in R . Also, for I an ideal of W ( R ) and a in R , let I ( a ) be the ideal of R formed by evaluating each element of I at a . We prove that if I and J are finitely generated ideals of W ( R ) such that I ( a ) = J ( a ) for all a in R then I = J (i.e., W ( R ) has the strong Skolem property).

Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

Let K be a field, and let A be a subring of K. We consider properties and applications of a compact, Hausdorff topology called the “ultrafilter topology” defined on the space Zar(K | A) of all valuation domains having K as quotient field and containing A. We show that the ultrafilter topology coincides with the constructible topology on the abstract Riemann-Zariski surface Zar(K | A). We extend results regarding distinguished spectral topologies on spaces of valuation domains.

CONSTRUCTING CHAINS OF PRIMES IN POWER SERIES RINGS, II

For an integral domain D of dimension n, the dimension of the polynomial ring D[x] is known to be bounded by n + 1 and 2n + 1. While n + 1 is a lower bound for the dimension of the power series ring D[[x]], it often happens that D[[x]] has infinite chains of primes. For example, such chains exist if D is either an almost Dedekind domain that is not Dedekind or a one-dimensional nondiscrete valuation domain. The main concern here is in developing a scheme by which such chains can be constructed in the gap between MV[[x]] and M[[x]] when V is a one-dimensional nondiscrete valuation domain with maximal ideal M. A consequence of these constructions is that there are chains of primes similar to the set of ω1 transfinite sequences of 0s and 1s ordered lexicographically.

Constructing Examples of Integral Domains by Intersecting Valuation Domains

The subject of this survey is a method of constructing integral domains which is not often utilized compared to various other methods, but is deceptively powerful. The motivation is the following classical theorem of W. Krull.

Some Closure Operations in Zariski-Riemann Spaces of Valuation Domains: A Survey

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains.