Summary: 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 \(\operatorname{Ass} (M/N)=\{ P_1, P_2, \dots , P_s\}\) and \(Q_i\) is a \(P_i\)-primary component of \(N \subsetneq M\) for each \(i=1,2,\dots,s\), then \(N =Q_1 \cap Q_2 \cap \cdots \cap Q_s\);
(2) for a given subset \(X=\{ P_1, P_2, \dots , P_r \} \subseteq \operatorname{Ass}(M/N)\), \(X\) is an open subset of \(\operatorname{Ass}(M/N)\) if and only if the intersections \(Q_1 \cap Q_2\cap \cdots \cap Q_r= Q_1' \cap Q_2' \cap \cdots \cap Q_r'\) 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 \(I_1, I_2, \dots, I_t\) of \(R\) there exists a \(k \in \mathbb N\) such that for any \(n_1, n_2, \dots, n_t \in \mathbb N\) there exists a primary decomposition of \(I_1^{n_1}I_2^{n_2}\cdots I_t^{n_t}M \subset M\) such that every \(P\)-primary component \(Q\) of that primary decomposition contains \(P^{k(n_1+n_2+\cdots+n_t)}M\).