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) ⇒ (b) ⇔ (c), and any ring R satisfying (c) has a right Morita duality.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm114-2-1,
author = {Nguyen Viet Dung and Jos\'e Luis Garc\'\i a},
title = {Rings whose modules are finitely generated over their endomorphism rings},
journal = {Colloquium Mathematicae},
volume = {116},
year = {2009},
pages = {155-176},
zbl = {1232.16003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm114-2-1}
}
Nguyen Viet Dung; José Luis García. Rings whose modules are finitely generated over their endomorphism rings. Colloquium Mathematicae, Tome 116 (2009) pp. 155-176. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm114-2-1/