×

Polynomials inducing the zero function on local rings. (English) Zbl 1369.13031

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 \(R/\mathfrak m\).

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
PDFBibTeX XMLCite
Full Text: DOI arXiv