Universal properties of integer-valued polynomial rings. (English) Zbl 1129.13022

Author’s abstract: Let \(D\) be an integral domain, and let \(A\) be a domain containing \(D\) with quotient field \(K\). We will say that the extension \(A\) of \(D\) is polynomially complete if \(D\) is a polynomially dense subset of \(A\), that is, if for all \(f\in K[X]\) with \(f(D)\subseteq A\) one has \(f(A)\subseteq A\). We show that, for any set \(\underline{X}\), the ring Int\((D^{\underline{X}})\) of integer-valued polynomials on \(D^{\underline{X}}\) is the free polynomially complete extension of \(D\) generated by \(\underline{X}\), provided only that \(D\) is not a finite field. We prove that a divisorial extension of a Krull domain \(D\) is polynomially complete if and only if it is unramified, and has trivial residue field extensions, at the height one primes in \(D\) with finite residue field. We also examine, for any extension \(A\) of a domain \(D\), the following three conditions:
(a) \(A\) is a polynomially complete extension of \(D\);
(b) Int\((A^{n})\supseteq\) Int\((D^{n})\) for every positive integer \(n\); and
(c) Int\((A)\supseteq\) Int\((D)\).
In general one has (a)\(\Rightarrow\)(b)\(\Rightarrow\)(c). It is known that (a)\(\Leftrightarrow\)(c) if \(D\) is a Dedekind domain. We prove various generalizations of this result, such as: (a)\(\Leftrightarrow\)(c) if \(D\) is a Krull domain and \(A\) is a divisorial extension of \(D\). Generally one has (b)\(\Leftrightarrow\)(c) if the canonical \(D\)-algebra homomorphism \(\varphi_{n}:\bigotimes_{i=1}^{n}\)Int\((D) \longrightarrow\) Int\((D^{n})\) is surjective for all positive integers \(n\), where the tensor product is over \(D\). Furthermore, \(\varphi_{n}\) is an isomorphism for all \(n\) if \(D\) is a Krull domain such that Int\((D)\) is flat as a \(D\)-module, or if \(D\) is a Prüfer domain such that Int\((D_{\mathcal{M}}) =\) Int\((D)_{\mathcal{M}}\) for every maximal ideal \(\mathcal{M}\) of \(D\).


13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13A15 Ideals and multiplicative ideal theory in commutative rings
Full Text: DOI


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