Rings whose modules are finitely generated over their endomorphism rings

Nguyen Viet Dung ¹   ; José Luis García ²

¹ Department of Mathematics Ohio University-Zanesville Zanesville, OH 43701, U.S.A.
² Department of Mathematics University of Murcia 30100 Espinardo, Murcia, Spain

Colloquium Mathematicum, Tome 114 (2009) no. 2, pp. 155-176

Résumé

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.

Détail

DOI : 10.4064/cm114-2-1

Keywords: module called finendo cofinendo finitely generated respectively finitely cogenerated its endomorphism ring proved hereditary ring following conditions equivalent nbsp every right r module finendo nbsp every r module cofinendo nbsp pure semisimple every finitely generated indecomposable r module cofinendo nbsp pure semisimple every finitely generated indecomposable r module finendo nbsp finite representation type moreover arbitrary ring rightarrow leftrightarrow ring satisfying has right morita duality