## Illinois Journal of Mathematics

### Big indecomposable modules and direct-sum relations

#### Abstract

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

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

Dates
First available in Project Euclid: 20 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258735327

Digital Object Identifier
doi:10.1215/ijm/1258735327

Mathematical Reviews number (MathSciNet)
MR2346189

Zentralblatt MATH identifier
1129.13010

#### Citation

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. https://projecteuclid.org/euclid.ijm/1258735327