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

Decomposition of integer-valued polynomial algebras