Illinois Journal of Mathematics

Big indecomposable modules and direct-sum relations

Wolfgang Hassler, Ryan Karr, Lee Klingler, and Roger Wiegand

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A commutative Noetherian local ring $(R,\m)$ is said to be \emph{Dedekind-like} provided $R$ has Krull-dimension one, $R$ has no non-zero nilpotent elements, the integral closure $\overline R$ of $R$ is generated by two elements as an $R$-module, and $\m$ is the Jacobson radical of $\overline R$. A classification theorem due to Klingler and Levy implies that if $M$ is a finitely generated indecomposable module over a Dedekind-like ring, then, for each minimal prime ideal $P$ of $R$, the vector space $M_P$ has dimension $0, 1$ or $2$ over the field $R_P$. The main theorem in the present paper states that if $R$ (commutative, Noetherian and local) has non-zero Krull dimension and is not a homomorphic image of a Dedekind-like ring, then there are indecomposable modules that are free of any prescribed rank at each minimal prime ideal.

Article information

Illinois J. Math., Volume 51, Number 1 (2007), 99-122.

First available in Project Euclid: 20 November 2009

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Zentralblatt MATH identifier

Primary: 13C13: Other special types
Secondary: 13D07: Homological functors on modules (Tor, Ext, etc.) 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations


Hassler, Wolfgang; Karr, Ryan; Klingler, Lee; Wiegand, Roger. Big indecomposable modules and direct-sum relations. Illinois J. Math. 51 (2007), no. 1, 99--122. doi:10.1215/ijm/1258735327.

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