Commutative rings and binomial coefficients
Copyright policy )Commutative rings and binomial coefficients
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 \(p\).
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510.mathematics, Integral domains, Noetherian domain, binomial coefficients, polynomial maps, Article, Polynomials in general fields (irreducibility, etc.), Polynomials over commutative rings
510.mathematics, Integral domains, Noetherian domain, binomial coefficients, polynomial maps, Article, Polynomials in general fields (irreducibility, etc.), Polynomials over commutative rings
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