Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains

Abstract

Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of integer-valued polynomials on $M$, $\text{Int}(M,D)=\{f\in K[x]|f(M)\subseteq D\}$. For every multiset of integers $1<z_1\le z_2\le \cdots \le z_n$, we explicitly construct an element of $\text{Int}(M,D)$ with exactly $n$ essentially different factorizations into irreducible elements of $\text{Int}(M,D)$ whose lengths are $z_1,z_2,\dots ,z_n$. Furthermore, we show that $\text{Int}(M,D)$ is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings $\text{Int}(S,D)\ne \text{Int}(D)$, where $S$ is an infinite subset of $D$.