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.
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