On the index of reducibility of parameter ideals: The stable and limit values

Let $(R, \frak m)$ 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 $M$.