On the index of reducibility of powers of a standard parameter ideal

Abstract

Throughout this paper, let (R, m) be a Noetherian local ring with the infinite residue field k = R/m, and M a finitely generalized R-module of dimension d. A submodule N of M is called an irreducible submodule if N can not be written as an intersection of two properly larger submodules of M. The number of irreducible components of an irredundant irreducible decomposition of N, which is independent of the choice of the decomposition by E. Noether [9], is called the index of reducibility of N, and denoted by irM (N). For a parameter ideal q of M, the index of ...