Let $E$ be the ring of integer valued polynomials over $\mathbb{Z}$. This ring is known to be a Prüfer domain. But it seems there does not exist an algorithm for inverting a nonzero finitely generated ideal of $E$. In this note we show how to obtain such an algorithm by deciphering a classical abstract proof that uses localisations of $E$ at all prime ideals of $E$. This confirms a general program of deciphering abstract classical proofs in order to obtain algorithmic proofs.

Let $R$ be an integral domain with quotient field $K$ and $f\left(x\right)$ a polynomial of positive degree in $K\left[x\right]$. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form $I=f\left(x\right)K\left[x\right]\cap R\left[x\right]$ are almost principal in the following two cases: – $J$, the ideal generated by the leading coefficients of $I$, satisfies $J^{-1}=R$. – $I^{-1}$ as the $R\left[x\right]$-submodule of $K\left(x\right)$ is of finite type. Furthermore we prove that for $I=f\left(x\right)K\left[x\right]\cap R\left[x\right]$ we have: – $I^{-1}\cap K\left[x\right]=\left(I{:}_{K\left(x\right)}I\right)$. – If there exists $p/q\in I^{-1}-K\left[x\right]$, then $(q,f)\ne 1$...