We show that if R is a ring such that every nonzero left R-module has a nonzero injective cover, then R is left Artinian. The converse is not true. If R is commutative, then the properties are equivalent.
Belshoff, Richard, and Jinzhong Xu. "Nonzero injective covers of modules." Missouri Journal of Mathematical Sciences 13, no. 3 (2001): 163-171.
Missouri Journal of Mathematical Sciences