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 ${\rm Int}(A) = {f \in B[X] \mid f(A) \subseteq A}$,andtheintersectionof${\rm Int}(A)$with$K[X]$is${\rm Int}_K(A)$,whichisacommutativesubringof$K[X]$\mathrm{.}Theset${\rm Int}(A)$mayormaynotbearing,butitalwayshasthestructureofaleft${\rm Int}_K(A)$-module. A $D$-algebra $A$ which is free as a $D$-module and of finite rank is called ${\mathrm{I}\mathrm{n}\mathrm{t}}_K$-decomposable if a $D$-module basis for $A$ is also an ${\rm Int}_K(A)$-modulebasisfor${\rm Int}(A)$;inotherwords,if${\rm Int}(A)$canbegeneratedby${\rm Int}_K(A)$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 ${\mathrm{I}\mathrm{n}\mathrm{t}}_K$-decomposable so that it can be applied to $D$-algebras that are not necessarily free by defining $A$ to be ${\mathrm{I}\mathrm{n}\mathrm{t}}_K$-decomposable when ${\rm Int}(A) \cong {\rm Int}_K(A) \otimes_D A$. 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 ${\mathrm{I}\mathrm{n}\mathrm{t}}_K$-decomposable algebras $A$ correspond to maximal $D$-orders 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 show that ${\mathrm{I}\mathrm{n}\mathrm{t}}_K$-decomposable algebras correspond to unramified Galois extensions of $K$.