Cofinitely semiperfect modules

It is well known that a projective module $M$ is $\oplus$-supplemented if and only if $M$ is semiperfect. We show that a projective module $M$ is $\oplus$-cofinitely supplemented if and only if $M$ is cofinitely semiperfect or briefly cof-semiperfect (i.e., each finitely generated factor module of $M$ has a projective cover). In this paper we give various properties of the cof-semiperfect modules. If a projective module $M$ is semiperfect then every $M$-generated module is cof-semiperfect. A ring $R$ is semiperfect if and only if every free $R$-module is cof-semiperfect.