## Rings whose modules are finitely generated over their endomorphism rings

### Volume 114 / 2009

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.