Rogers, Mark W.; Wickham, Cameron Polynomials inducing the zero function on local rings. (English) Zbl 1369.13031 Int. Electron. J. Algebra 22, 170-186 (2017). For a commutative ring \(R\) let \(N(R)\) be the ideal of \(R[X]\) consisting of all polynomials vanishing on \(R\). The author considers the case when \(R\) is a Noetherian local ring with prime ideal \(\mathfrak m\), and studies the ideals \(N(R)\) and \(N(\mathfrak m)\). It is shown (Theorem 3.3) that if \(R\) is finite, then \(N(R)\) is principal if and only if \(R\) is a field, and the same holds for \(N(\mathfrak m)\). The corollary to the next result describes the case when \(N(R)\) or \(N(\mathfrak m)\) contains (or is generated by) regular polynomials (Corollary 3.5), and in the main result of the next section (Theorem 4.2) the author establishes that if \(R\) is a Henselian local ring with finite residue field, and \(N(\mathfrak m)\) is generated by \(F_1,\dots,F_n\), then \(N(R)\) is generate by \(F_1\circ \pi,\dots,F_n\circ\pi\), where \(\pi(X)=\prod_i(X-c_i)\) with \(\{c_i\}\) being a set of representatives of \(R/\mathfrak m\). Reviewer: Władysław Narkiewicz (Wrocław) Cited in 1 ReviewCited in 5 Documents MSC: 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 11C08 Polynomials in number theory 13B25 Polynomials over commutative rings 13E10 Commutative Artinian rings and modules, finite-dimensional algebras 13H99 Local rings and semilocal rings 13J15 Henselian rings 13M10 Polynomials and finite commutative rings Keywords:null ideal; finite ring; vanishing polynomial; Artinian ring PDFBibTeX XMLCite \textit{M. W. Rogers} and \textit{C. Wickham}, Int. Electron. J. Algebra 22, 170--186 (2017; Zbl 1369.13031) Full Text: DOI arXiv