Abstract
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 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 an = qnln... fnqnkn with +/hn C qni for all i = 1,... , kn. The generalization involves a finitely generated R-module and several ideals; the short proof is based on the theory of injective R-modules.
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