Holomorphy rings of function fields

Abstract

In his 1974 text, Commutative Ring Theory, Kaplansky states that among the examples of non-Dedekind Priifer domains, the main ones are valuation domains, the ring of entire functions and the integral closure of a Priifer domain in an algebraic extension of its quotient field [Kap74, p.72]. A similar list today would likely include Kronecker function rings, the ring of integervalued polynomials and real holomorphy rings. All of these examples of Priifer domains have been fundamental to the development of multiplicative ideal theory, as is evidenced in the work of Robert Gilmer over the past 40 years. These rings have been intensely studied from various points of views and with different motivations and tools. In this article we make some observations regarding the ideal theory of holomorphy rings of function fields.