Let M be a left module over a ring R. M is called a Zelmanowitz-regular module if for each x ∈ M there exists a homomorphism F: M → R such that f(x) = x. Let Q be a left R-module and h: Q → M a homomorphism. We call h locally split if for every x ∈ M there exists a homomorphism g: M → Q such that h(g(x)) = x. M is called locally projective if every epimorphism onto M is locally split. We prove that the following conditions are equivalent:
(1) M is Zelmanowitz-regular.
(2) every homomorphism into M is locally split.
(3) M is locally projective and every cyclic submodule of M is a direct summand of M.
@article{urn:eudml:doc:41129,
title = {Some characterizations of regular modules.},
journal = {Publicacions Matem\`atiques},
volume = {34},
year = {1990},
pages = {241-248},
zbl = {0722.16001},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41129}
}
Azumaya, Goro. Some characterizations of regular modules.. Publicacions Matemàtiques, Tome 34 (1990) pp. 241-248. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41129/