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 is an element of B[X] vertical bar f(A) subset of 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) circle times(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. (C) 2017 Elsevier B.V. All rights reserved.