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Injective modules and linear growth of primary decompositions

Authors:	R Y Sharp

Institution:	Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Abstract:	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 has linear growth of primary decompositions, that is, there exists a positive integer such that, for every positive integer , 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 -module and several ideals; the short proof is based on the theory of injective -modules.

Keywords:	Commutative Noetherian ring primary decomposition associated prime ideal injective module Artin-Rees Lemma

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