Primary decomposition II: Primary components and linear growth

We study the following properties about primary decomposition over a Noetherian ring R: (1) For finitely generated modules N⊆M and a given subset X={P₁,P₂,…,P_r}⊆Ass(M/N), we define an X-primary component of N?M to be an intersection Q₁∩Q₂∩?∩Q_r for some P_i-primary components Q_i of N⊆M and we study the maximal X-primary components of N⊆M; (2) We give a proof of the ‘linear growth’ property of Ext and Tor, which says that for finitely generated modules N and M, any fixed ideals I₁,I₂,…,I_t of R and any fixed integer i∈N, there exists a k∈N such that for any there exists a primary decomposition of 0 in (or 0 in ) such that every P-primary component Q of that primary decomposition contains (or ), where .