Decomposition of Integer-valued Polynomial Algebras

Let $D$ be a commutative domain with field of fractions $K$, let $A$ be a torsion-free $D$-algebra, and let $B$ be the extension of $A$ to a $K$-algebra. The set of integer-valued polynomials on $A$ is $Int(A) = {f in B[X] mid f(A) subseteq A}$, and the intersection of $Int(A)$ with $K[X]$ is $Int_K(A)$, which is a commutative subring of $K[X]$. The set $Int(A)$ may or may not be a ring, but it always has the structure of a left $Int_K(A)$-module. A $D$-algebra $A$ which is free as a $D$-module and of finite rank is called $Int_K$-decomposable if a $D$-module basis for $A$ is also an $Int_K(A)$-module basis for $Int(A)$; in other words, if $Int(A)$ can be generated by $Int_K(A)$ and $A$. A classification of such algebras has been given when $D$ is a Dedekind domain with finite residue rings. In the present article, we modify the definition of $Int_K$-decomposable so that it can be applied to $D$-algebras that are not necessarily free by defining $A$ to be $Int_K$-decomposable when $Int(A)$ is isomorphic to $Int_K(A) otimes_D A$. We then provide multiple characterizations of such algebras in the case where $D$ is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if $D$ is the ring of integers of a number field $K$, we show that an $Int_K$-decomposable algebra $A$ must be a maximal $D$-order in a separable $K$-algebra $B$, whose simple components have as center the same finite unramified Galois extension $F$ of $K$ and are unramified at each finite place of $F$. Finally, when both $D$ and $A$ are rings of integers in number fields, we prove that $Int_K$-decomposable algebras correspond to unramified Galois extensions of $K$