Holomorphy rings of function fields

In his 1974 text, Commutative Ring Theory, Kaplansky states that among the examples of non-Dedekind Prüfer domains, the main ones are valuation domains, the ring of entire functions and the integral closure of a Prüfer 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 Prüfer 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. A holomorphy ring is an intersection of valuation rings having a common quotient field F . The terminology arises from viewing elements of F as functions on collections of valuation rings having quotient field F . To formulate this more precisely, let F be a field and D be a subring of F . The Zariski-Riemann space of F is the collection Σ(F |D) of all valuation rings V containing D and having quotient field F . If D is the prime subring of F , then we write Σ(F ) for Σ(F |D). One can introduce a topology on Σ(F |D) in a natural way [ZS75, p. 110]. In Section 2 we will consider the Zariski patch topology on Σ(F |D). If S ⊆ Σ(F |D), then x ∈ F is holomorphic on S if x has no pole on S. More precisely, for each V ∈ S, let φV : F → FV ∪ {∞} be the place corresponding to V , where FV is the residue field of V . Then x assigns to V the value φV (x). Thus x is holomorphic on S if and only if x is finite on each