On the number of ideals of finite colength (Q2717159)

This paper deals with Noetherian rings in which there is a finite set of ideals with finite colength. Such a ring is semilocal and each localization at a maximal ideal has the same property. It is shown that in a local ring \(R\) with \(\dim R \geq 2\), the number \(\Omega(h)\) of ideals of colength \(h\) grows exponentially with \(h\). If \(R\) is a one-dimensional analytically unramified residually rational local ring with finite residue field of characteristics \(\geq d\), where \(d\) is the number of the maximal ideals of the integral closure of \(R\), then \(\Omega(h)\) is a polynomial of degree \(d-1\) for \(h \geq \ell(\overline R/R:\overline R)\). The proof uses the value semigroup associated to such a ring investigated by \textit{V. Barucci, M. D'Anna} and \textit{R. Fröberg} [J. Pure Appl. Algebra 147, No. 3, 215-254 (2000; Zbl 0963.13021)].NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].