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Krull Dimension of Injective Modules Over Commutative Noetherian Rings

Published online by Cambridge University Press:  20 November 2018

Patrick F. Smith
Patrick F. Smith*
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland, U.K.
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Abstract

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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.

Keywords

13E0516D5016P60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 2 , 01 June 2005 , pp. 275 - 282
Copyright
Copyright © Canadian Mathematical Society 2005

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