On the index of reducibility in Noetherian modules

Abstract

Abstract Let M be a finitely generated module over a Noetherian ring R and N be a submodule. The index of reducibility ir M (N) ir M (N) is the number of irreducible submodules that appear in an irredundant irreducible decomposition of N (this number is well defined by a classical result of Emmy Noether). Then the main results of this paper are:(1) ir M (N)=∑ p∈ Ass R (M/N) dim k (p)⁡ Soc (M/N) p ir M (N)=∑ p∈ Ass R (M/N) dim k (p)⁡ Soc (M/N) p;(2) For an irredundant primary decomposition of N= Q 1∩⋯∩ Q n N= Q ...