Integer-valued polynomials on Krull rings

If $R$ is a subring of a Krull ring $S$ such that $R_{Q}$ is a valuation ring for every finite index $Q=P\cap R$, $P$ in Spec$^{1}(S)$, we construct polynomials that map $R$ into the maximal possible (for a monic polynomial of fixed degree) power of $PS_{P}$, for all $P$ in Spec$^{1}(S)$ simultaneously. This gives a direct sum decomposition of Int$(R,S)$, the $S$-module of polynomials with coefficients in the quotient field of $S$ that map $R$ into $S$, and a criterion when Int$(R,S)$ has a regular basis (one consisting of 1 polynomial of each non-negative degree).