Commutative rings and binomial coefficients (Q1207655)

Let \(R\) be a domain with quotient field \(k\) and let \(S(R)\) be the \(R\)- module of all \(k\)-polynomials mapping \(R\) in \(R\). It has been shown by \textit{G. Pólya} [Rend. Circ. Mat. Palermo 40, 1-16 (1915; JFM 45.0655.02)] that \(S(Z)\) is generated by binomial coefficients. \textit{G. Gerboud} [C. R. Acad. Sci., Paris, Sér. A 307, 1-4 (1988; Zbl 0656.13022)] found other such domains. Here all domains \(R\) with this property are described. In particular a Noetherian domain \(R\) has this property if and only if for every rational prime \(p\) non-invertible in \(R\) the ideal \(pR\) is a product of distinct prime ideals of index \(p\).