If M is a simple module over a ring R then, by the Schur's lemma, the endomorphism ring of M is a division ring. However, the converse of this result does not hold in general, even when R is artinian. In this short note, we consider perfect rings for which the converse assertion is true, and we show that these rings are exactly the primary decomposable ones.
@article{urn:eudml:doc:41426,
title = {Perfect rings for which the converse of Schur's lemma holds.},
journal = {Publicacions Matem\`atiques},
volume = {45},
year = {2001},
pages = {219-222},
mrnumber = {MR1829585},
zbl = {0982.16003},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41426}
}
Haily, Abdelfattah; Alaoui, Mostafa. Perfect rings for which the converse of Schur's lemma holds.. Publicacions Matemàtiques, Tome 45 (2001) pp. 219-222. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41426/