Halter-Koch, F.; Narkiewicz, W. Commutative rings and binomial coefficients. (English) Zbl 0764.12002 Monatsh. Math. 114, No. 2, 107-110 (1992). 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 G. Pólya [Rend. Circ. Mat. Palermo 40, 1-16 (1915; JFM 45.0655.02)] that \(S(Z)\) is generated by binomial coefficients. 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 \(p\). Reviewer: F.Halter-Koch (Graz) Cited in 1 ReviewCited in 2 Documents MSC: 12E05 Polynomials in general fields (irreducibility, etc.) 13G05 Integral domains 13B25 Polynomials over commutative rings Keywords:polynomial maps; binomial coefficients; Noetherian domain Citations:Zbl 0656.13022; JFM 45.0655.02 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Gerboud, G.: Exemples d’anneauxA pour lesquels \(((\begin{array}{*{20}c} X n \end{array} ))_{n \in N} \) est une base duA-module des polynomes enti?res surA. Comptes Rendus Acad. Sci. Paris307, 1-4 (1988). [2] P?lya, G.: ?ber ganzwertige ganze Funktionen. Rendiconti Circ. Mat. Palermo40, 1-16 (1915). · JFM 45.0655.02 · doi:10.1007/BF03014836 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.