Ring properties of endomorphism rings of modules

A certain method of studying ring properties of endomorphism rings of modules is justified. As an example of its applications the equivalence of the following conditions is proved: 1) the right annihilator of every proper finitely generated (principal) left ideal in any endomorphism ring of an injective right $R$-module contains a nonzero idempotent; 2) the ring $R$ is a semiartinian right $V$-ring.