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Direct-sum behavior of modules over one-dimensional rings

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Commutative Algebra

Abstract

Let R be a reduced, one-dimensional Noetherian local ring whose integral closure \(\overline{R}\) is finitely generated over R. Since \(\overline{R}\) is a direct product of finitely many principal ideal domains (one for each minimal prime ideal of R), the indecomposable finitely generated \(\overline{R}\)-modules are easily described, and every finitely generated \(\overline{R}\)-module is uniquely a direct sum of indecomposable modules. In this article we will see how little of this good behavior trickles down to R. Indeed, there are relatively few situations where one can describe all of the indecomposable R-modules, or even the torsion-free ones. Moreover, a given finitely generated module can have many different representations as a direct sum of indecomposable modules.

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816-1364, USA

    Ryan Karr

  2. Department of Mathematics, University of Nebraska, Lincoln, NE, 68588-0130, USA

    Roger Wiegand

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  1. Ryan Karr
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  2. Roger Wiegand
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Correspondence to Ryan Karr .

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Editors and Affiliations

  1. , Dipartimento di Matematica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo 1, Roma, 00146, Italy

    Marco Fontana

  2. , Department of Mathematics and Statistics, King Fahd University of Petroleum & Mine, Dhahran, 31261, Saudi Arabia

    Salah-Eddine Kabbaj

  3. , Department of Mathematical Sciences, New Mexico State University, Las Cruces, 88003, New Mexico, USA

    Bruce Olberding

  4. , Department of Mathematics, Reed College, Southeast Woodstock Blvd. 3203, Portland, 97202-8199, Oregon, USA

    Irena Swanson

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Karr, R., Wiegand, R. (2011). Direct-sum behavior of modules over one-dimensional rings. In: Fontana, M., Kabbaj, SE., Olberding, B., Swanson, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6990-3_10

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