Primary Decomposition: Compatibility, Independence and Linear Growth

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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 (M/N)={P₁, P₂,..., Pₛ} and Qᵢ is a Pᵢ-primary component of N M for each i=1,2,...,s, then N =Q₁ ∩ Q₂ ∩... ∩ Qₛ; (2) For a given subset X={P₁, P₂,..., Pᵣ } (M/N), X is an open subset of (M/N) if and only if the intersections Q₁ ∩ Q₂∩... ∩ Qᵣ= Q₁' ∩ Q₂' ∩... ∩ Qᵣ' for all possible Pᵢ-primary components Qᵢ and Qᵢ' of N M; (3) A new proof of the `Linear Growth' property, which says that for any fixed ideals I₁, I₂,..., Iₜ of R, there exists a k ∈ N such that for any n₁, n₂,..., nₜ ∈ N there exists a primary decomposition of I₁ⁿ¹I₂ⁿ²... IₜⁿᵗM ⊂ M such that every P-primary component Q of that primary decomposition contains Pᵏ⁽ⁿ¹⁺ⁿ²⁺...⁺ⁿᵗ⁾M.