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 a in a commutative Noetheri an ring R has linear growth of primary decompositions, that is, there exist s a positive integer h such that, for every positive integer n, there exist s a minimal primary decomposition a(n) = q(n1) boolean AND...boolean AND q( nkn) with root q(ni)(hn) subset of or equal to q(ni) for all i = 1, ..., k( n). The generalization involves a finitely generated R-module and several i deals; the short proof is based on the theory of injective R-modules.