Injective modules and linear growth of primary decompositions

The purposes of this paper are to generalize, and to provide a short proof of, I. Swanson's Theorem that each proper ideal $\mathfrak{a}$ in a commutative Noetherian ring $R$ has linear growth of primary decompositions, that is, there exists a positive integer $h$ such that, for every positive integer $n$, there exists a minimal primary decomposition ${\mathfrak{a}}^{n} = {\mathfrak{q}}_{n1} \cap \ldots \cap {\mathfrak{q}}_{nk_{n}}$ with $\sqrt {{\mathfrak{q}}_{ni}}^{hn} \subseteq {\mathfrak{q}}_{ni}$ for all $i =1, \ldots , k_{n}$. The generalization involves a finitely generated $R$-module and several ideals; the short proof is based on the theory of injective $R$-modules.