Geodesic
Krull Dimension of Injective Modules Over Commutative Noetherian Rings
Canadian mathematical bulletin, Tome 48 (2005) no. 2, pp. 275-282

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DOI

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.
DOI : 10.4153/CMB-2005-026-3
Mots-clés : 13E05, 16D50, 16P60 
Smith, Patrick F. Krull Dimension of Injective Modules Over Commutative Noetherian Rings. Canadian mathematical bulletin, Tome 48 (2005) no. 2, pp. 275-282. doi: 10.4153/CMB-2005-026-3
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     author = {Smith, Patrick F.},
     title = {Krull {Dimension} of {Injective} {Modules} {Over} {Commutative} {Noetherian} {Rings}},
     journal = {Canadian mathematical bulletin},
     pages = {275--282},
     year = {2005},
     volume = {48},
     number = {2},
     doi = {10.4153/CMB-2005-026-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2005-026-3/}
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