On the endofiniteness of a key module over pure semisimple rings

Dung, Nguyen; García, José

doi:10.1090/S0002-9939-10-10098-7

On the endofiniteness of a key module over pure semisimple rings
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by Nguyen Viet Dung and José Luis García PDF

Proc. Amer. Math. Soc. 138 (2010), 2269-2278 Request permission

Abstract:

Let $R$ be a left pure semisimple ring such that there are no non-zero homomorphisms from preinjective modules to non-preinjective indecomposable modules in $R$-mod, and let $W$ be the left key $R$-module; i.e., $W$ is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left $R$-modules. We show that if the module $W$ is endofinite, then $R$ is a ring of finite representation type. This settles a question considered in [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007), 361-376] for hereditary rings.

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