Rings whose modules are finitely generated over their endomorphism rings

Volume 114 / 2009

Nguyen Viet Dung, José Luis García Colloquium Mathematicum 114 (2009), 155-176 MSC: 16G10, 16D70, 16D90. DOI: 10.4064/cm114-2-1

Abstract

A module $M$ is called finendo (cofinendo) if $M$ is finitely generated (respectively, finitely cogenerated) over its endomorphism ring. It is proved that if $R$ is any hereditary ring, then the following conditions are equivalent: (a) Every right $R$-module is finendo; (b) Every left $R$-module is cofinendo; (c) $R$ is left pure semisimple and every finitely generated indecomposable left $R$-module is cofinendo; (d) $R$ is left pure semisimple and every finitely generated indecomposable left $R$-module is finendo; (e) $R$ is of finite representation type. Moreover, if $R$ is an arbitrary ring, then (a)$\Rightarrow $(b)$\Leftrightarrow $(c), and any ring $R$ satisfying (c) has a right Morita duality.

Authors

  • Nguyen Viet DungDepartment of Mathematics
    Ohio University-Zanesville
    Zanesville, OH 43701, U.S.A.
    e-mail
  • José Luis GarcíaDepartment of Mathematics
    University of Murcia
    30100 Espinardo, Murcia, Spain
    e-mail

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