On the index of reducibility in Noetherian modules

Let M be a finitely generated module over a Noetherian ring R and N   be a submodule. The index of reducibility irM(N)irM(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) irM(N)=∑p∈AssR(M/N)dimk(p)⁡Soc(M/N)pirM(N)=∑p∈AssR(M/N)dimk(p)⁡Soc(M/N)p; (2) For an irredundant primary decomposition of N=Q1∩⋯∩QnN=Q1∩⋯∩Qn, where QiQi is pipi-primary, irM(N)=irM(Q1)+⋯+irM(Qn)irM(N)=irM(Q1)+⋯+irM(Qn) if and only if QiQi is a pipi-maximal embedded component of N   for all embedded associated prime ideals pipi of N; (3) For an ideal I of R   there exists a polynomial IrM,I(n)IrM,I(n) such that IrM,I(n)=irM(InM)IrM,I(n)=irM(InM) for n≫0n≫0. Moreover, bightM(I)−1≤deg⁡(IrM,I(n))≤ℓM(I)−1bightM(I)−1≤deg⁡(IrM,I(n))≤ℓM(I)−1; (4) If (R,m)(R,m) is local, M is Cohen–Macaulay if and only if there exist an integer l   and a parameter ideal qq of M   contained in mlml such that irM(qM)=dimR/m⁡Soc(Hmd(M)), where d=dim⁡Md=dim⁡M.