Let Λ be an artinian ring and let 𝔯 denote its Jacobson radical. We show that a simple module of finite projective dimension has no self-extensions when Λ is graded by its radical, with at most two simple modules and 𝔯⁴ = 0, in particular, when Λ is a finite-dimensional algebra over an algebraically closed field with at most two simple modules and 𝔯³ = 0.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm102-1-1,
author = {Bernt T. Jensen},
title = {Strong no-loop conjecture for algebras with two simples and radical cube zero},
journal = {Colloquium Mathematicae},
volume = {103},
year = {2005},
pages = {1-7},
zbl = {1092.16005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm102-1-1}
}
Bernt T. Jensen. Strong no-loop conjecture for algebras with two simples and radical cube zero. Colloquium Mathematicae, Tome 103 (2005) pp. 1-7. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm102-1-1/