In general, the injective cover (projective envelope) of a simple module can be zero. A ring R is called a weakly left V-ring (strongly left Kasch ring) if every simple left R-module has a nonzero injective cover (projective envelope). It is proven that every nonzero left R-module has a nonzero injective cover if and only if R is a left Artinian weakly left V-ring. Dually, every nonzero left R-module has a nonzero projective envelope if and only if R is a left perfect right coherent strongly left Kasch ring. Some related rings and examples are considered.
Injective cover, projective envelope, weakly V-ring, strongly Kasch ring
ZHANG, XIAOXIANG and SONG, XIANMEI
"On the existence of nonzero injective covers and projective envelopes of modules,"
Turkish Journal of Mathematics: Vol. 37:
6, Article 3.
Available at: https://journals.tubitak.gov.tr/math/vol37/iss6/3