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Krull Dimension of Injective Modules Over Commutative Noetherian Rings
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $R$ be a commutative Noetherian integral domain with field of fractions
$Q$. Generalizing a forty-year-old theorem of E. Matlis, we prove that the
$R$-module
$Q/R$ (or
$Q$) has Krull dimension if and only if
$R$ is semilocal and one-dimensional. Moreover, if
$X$ is an injective module over a commutative Noetherian ring such that
$X$ has Krull dimension, then the Krull dimension of
$X$ is at most 1.
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- Copyright © Canadian Mathematical Society 2005