On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values

Abstract

Let ( R , m ) $(R, \\frak {m})$ be a Noetherian local ring and M a finitely generated R -module of dimension d . A famous result of Northcott says that if M is Cohen-Macaulay, then the index of reducibility of parameter ideals on M is an invariant of the module. The aim of this paper is to extend Northcott’s theorem for any finitely generated R -module. We call this invariant the stable value of the indices of reducibility of parameter ideals of M . We also introduce the limit value of the indices of reducibility of parameter ideals of M .