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Birings and plethories of integer-valued polynomials. (English) Zbl 1439.13062

Summary: Let \(A\) and \(B\) be commutative rings with identity. An \(A\)-\(B\)-biring is an \(A\)-algebra \(S\) together with a lift of the functor \(\operatorname{Hom}_A(S, -)\) from \(A\)-algebras to sets to a functor from \(A\)-algebras to \(B\)-algebras. An \(A\)-plethory is a monoid object in the monoidal category, equipped with the composition product, of \(A\)-\(A\)-birings. The polynomial ring \(A [X]\) is an initial object in the category of such structures. The \(D\)-algebra \(\operatorname{Int}(D)\) has such a structure if \(D = A\) is a domain such that the natural \(D\)-algebra homomorphism \(\theta_n : \bigotimes_{D_{i = 1}}^n \operatorname{Int}(D) \rightarrow \operatorname{Int}(D^n)\) is an isomorphism for \(n = 2\) and injective for \(n \leq 4\). This holds in particular if \(\theta_n\) is an isomorphism for all \(n\), which in turn holds, for example, if \(D\) is a Krull domain or more generally a TV PVMD. In these cases we also examine properties of the functor \(\operatorname{Hom}_D(\operatorname{Int}(D), -)\) from \(D\)-algebras to \(D\)-algebras, which we hope to show is a new object worthy of investigation in the theory of integer-valued polynomials.

MSC:

13G05 Integral domains
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
16W99 Associative rings and algebras with additional structure

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