Krull Dimension of Injective Modules Over Commutative Noetherian Rings

http://dx.doi.org/10.4153/CMB-2005-026-3
Canad. Math. Bull. 48(2005), 275-282

  Published:2005-06-01
 Printed: Jun 2005

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 $1$.