On the index of reducibility in Noetherian modules

Let $M$ be a finitely generated module over a Noetherian ring $R$ and $N$ a submodule. The index of reducibility 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) $$\mathrm{i}\mathrm{r}$M(N) = \sum{{\frak p} \in $\mathrm{A}\mathrm{s}\mathrm{s}$R(M/N)} \dim{k(\frak p)} $\mathrm{S}\mathrm{o}\mathrm{c}$(M/N)_{\frak p} $; (2) For an irredundant primary decomposition of $N=Q_1\cap \cdots \cap Q_n$, where $Q_i$ is ${\mathfrak{p}}_i$-primary, then $$\mathrm{i}\mathrm{r}$_M(N) = $\mathrm{i}\mathrm{r}$_M(Q_1) + \cdots + $\mathrm{i}\mathrm{r}$_M(Q_n)$ifandonlyif$Q_i$isa$\frak p_i$-maximal embedded component of $N$ for all embedded associated prime ideals ${\mathfrak{p}}_i$ of $N$; (3) For an ideal $I$ of $R$ there exists a polynomial ${\mathrm{I}\mathrm{r}}_{M,I}(n)$ such that ${\mathrm{I}\mathrm{r}}_{M,I}(n)={\mathrm{i}\mathrm{r}}_M(I^{n}M)$ for $n\gg 0$. Moreover, ${\mathrm{b}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}_M(I)-1\le \mathrm{deg}({\mathrm{I}\mathrm{r}}_{M,I}(n))\le {\mathrm{\ell }}_M(I)-1$; (4) If $(R,\mathfrak{m})$ is local, $M$ is Cohen-Macaulay if and only if there exist an integer $l$ and a parameter ideal $\mathfrak{q}$ of $M$ contained in ${\mathfrak{m}}^l$ such that ${\mathrm{i}\mathrm{r}}_M(\mathfrak{q}M)={\dim }_k\mathrm{S}\mathrm{o}\mathrm{c}(H_{\mathfrak{m}}^d(M))$, where $d=\dim M$.