Decomposition of integer-valued polynomial algebras

Abstract

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 $\mathrm{Int}(A)=\{f\in B[X]\mid f(A)\subseteq A\}$, and the intersection of $\mathrm{Int}(A)$ with $K[X]$ is ${\mathrm{Int}}_K(A)$, which is a commutative subring of $K[X]$. The set $\mathrm{Int}(A)$ may or may not be a ring, but it always has the structure of a left ${\mathrm{Int}}_K(A)$-module. A $D$-algebra $A$ which is free as a $D$-module and of finite rank is called ${\mathrm{Int}}_K$-decomposable if a $D$-module basis for $A$ is also an ${\mathrm{Int}}_K(A)$-module basis for $\mathrm{Int}(A)$; in other words, if $\mathrm{Int}(A)$ can be generated by ${\mathrm{Int}}_K(A)$ 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 ${\mathrm{Int}}_K$-decomposable so that it can be applied to $D$-algebras that are not necessarily free by defining $A$ to be ${\mathrm{Int}}_K$-decomposable when $\mathrm{Int}(A)\cong {\mathrm{Int}}_K(A){\otimes }_{D}A$. 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 ${\mathrm{Int}}_K$-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 ${\mathrm{Int}}_K$-decomposable algebras correspond to unramified Galois extensions of $K$.

to appear in J. Pure Appl. Algebra (2017). comments are welcome

References in corpus (6)

• Properly Integral Polynomials over the Ring of Integer-valued Polynomials on a Matrix Ring
• Integral closure of rings of integer-valued polynomials on algebras
• The ring of polynomials integral-valued over a finite set of integral elements
• Polynomial overrings of $\mathrm{Int}(\mathbb{Z})$
• Non-triviality Conditions for Integer-valued Polynomial Rings on Algebras
• Galois structure on integral valued polynomials

Cited by in corpus (1)

• Polynomial functions on non-commutative rings - a link between ringsets and null-ideal sets