## One-dimensional rings of finite Cohen-Macaulay type

### Abstract

Let R be a one-dimensional local Noetherian ring. A non-zero R-module M is said to be a maximal Cohen-Macaulay module if it is finitely generated, and the unique maximal ideal of R contains a non-zero-divisor on M. The Main Theorem (1.5) states that for a one-dimensional local Cohen-Macaulay ring R, the following conditions are necessary and sufficient for R to have only finitely many indecomposable finitely generated maximal Cohen-Macaulay modules up to isomorphism: (1) R is reduced; (2) the integral closure $\tilde R$ of R in its total quotient ring can be generated by 3 elements as an R-module; and (3) the intersection of the maximal R-submodules of $\tilde R$/R is a cyclic R-module

Topics: Mathematics
Publisher: DigitalCommons@University of Nebraska - Lincoln
Year: 1994
OAI identifier: oai:digitalcommons.unl.edu:dissertations-7779
Provided by: UNL | Libraries