Primary Decomposition: Compatibility, Independence and Linear Growth

For finitely generated modules $N⊊M$ over a Noetherian ring $R$, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if $\text{ass}(M\mathrm{/}N)=\{P_1,P_2,\mathrm{.}\mathrm{.}\mathrm{.},P_s\}$ and $Q_i$ is a $P_i$-primary component of $N⊊M$ for each $i=1,2,\mathrm{.}\mathrm{.}\mathrm{.},s$, then $N=Q_1\cap Q_2\cap \mathrm{.}\mathrm{.}\mathrm{.}\cap Q_s$; (2) For a given subset $X=\{P_1,P_2,\mathrm{.}\mathrm{.}\mathrm{.},P_r\}\subseteq \text{ass}(M\mathrm{/}N)$, $X$ is an open subset of $\text{ass}(M\mathrm{/}N)$ if and only if the intersections $Q_1 \cap Q_2\cap … \cap Q_r= Q_1’ \cap Q_2’ \cap … \cap Q_r’$forallpossible$P_i$-primarycomponents$Q_i$and$Q_i’$of$N\subsetneq M$; (3) A new proof of the `Linear Growth’ property, which says that for any fixed ideals $I_1,I_2,\mathrm{.}\mathrm{.}\mathrm{.},I_t$ of $R$, there exists a $k\in \mathbb{N}$ such that for any $n_1, n_2, …, n_t \in \mathbb N$thereexistsaprimarydecompositionof$I_1^{n_1}I_2^{n_2}… I_t^{n_t}M \subset M$suchthatevery$P$-primarycomponent$Q$ of that primary decomposition contains $P^{k(n_1+n_2+\mathrm{.}\mathrm{.}\mathrm{.}+n_t)}M$.