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# Krull Dimension of Injective Modules Over Commutative Noetherian Rings

Published:2005-06-01
Printed: Jun 2005
• Patrick F. Smith
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## 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 $1$.
 MSC Classifications: 13E05 - Noetherian rings and modules 16D50 - Injective modules, self-injective rings [See also 16L60] 16P60 - Chain conditions on annihilators and summands: Goldie-type conditions [See also 16U20], Krull dimension