Birings and plethories of integer-valued polynomials

Abstract

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 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 Int(D) has such a structure if D=A is a domain such that the natural D-algebra homomorphism θ n :⨂ D i=1 n Int(D)→Int(D n ) is an isomorphism for n=2 and injective for n≤4. This holds in particular if θ 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 Hom D (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.