On the existence of nonzero injective covers and projective envelopes of modules

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.