Primary Decomposition: Compatibility, Independence and Linear Growth

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