Essential properties for rings of integer-valued polynomials. (English) Zbl 1522.13005
Let \(D\) be an integral domain with quotient field \(K\) and \(\operatorname{Int}(D) := \{f \in K[X] \mid f(D) \subseteq D \}\) be the ring of integer-valued polynomials over \(D\). It is known that when \(\operatorname{Int}(D)\) is a Krull domain [P.-J. Cahen et al., J. Pure Appl. Algebra 153, No. 1, 1–15 (2000; Zbl 0978.13010), Corollary 2.7] and a Prüfer \(v\)-multiplication domain (P\(v\)MD for short) [P.-J. Cahen et al., J. Algebra 226, No. 2, 765–787 (2000; Zbl 0961.13012), Theorem 3.4]. Motivated by these results, this paper investigates when \(\operatorname{Int}(D)\) has essential-type properties. In particular, the authors give a complete characterization of when \(\operatorname{Int}(D)\) is locally essential, locally P\(v\)MD, locally UFD, locally GCD, Krull-type, or generalized Krull.
Reviewer: Hwankoo Kim (Asan)
MSC:
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13A18 | Valuations and their generalizations for commutative rings |
| 13B30 | Rings of fractions and localization for commutative rings |
| 13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
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