On the index of reducibility in Noetherian modules

Abstract

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) ${\mathrm{ir}}_M(N)={\sum }_{\mathfrak{p}\in {\mathrm{Ass}}_R(M/N)}{\dim }_{k(\mathfrak{p})}\mathrm{Soc}(M/N{)}_{\mathfrak{p}}$; (2) For an irredundant primary decomposition of $N=Q_1\cap \cdots \cap Q_n$, where $Q_i$ is ${\mathfrak{p}}_i$-primary, then ${\mathrm{ir}}_M(N)={\mathrm{ir}}_M(Q_1)+\cdots +{\mathrm{ir}}_M(Q_n)$ if and only if $Q_i$ is a ${\mathfrak{p}}_i$-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 ${\mathrm{Ir}}_{M,I}(n)$ such that ${\mathrm{Ir}}_{M,I}(n)={\mathrm{ir}}_M(I^{n}M)$ for $n\gg 0$. Moreover, ${\mathrm{bight}}_M(I)-1\le °({\mathrm{Ir}}_{M,I}(n))\le {\ell }_M(I)-1$; (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 ${\mathrm{ir}}_M(\mathfrak{q}M)={\dim }_k\mathrm{Soc}(H_{\mathfrak{m}}^d(M))$, where $d=\dim M$.

14 pages, To appear in J. Pure Appl. Algebra

References in corpus (1)

• Primary Decomposition: Compatibility, Independence and Linear Growth