A generalization of projective covers
The authors generalize the notion of a projective cover of a module to the projective \(I\)-cover of a module for an ideal \(I\) of a ring \(R\). They prove that a ring \(R\) is \(I\)-semiregular if and only if every cyclically presented \(R\)-module has a projective \(I\)-cover. Using this they derive equivalent conditions for a ring \(R\) to be \(Z(_RR)\)-semiregular and \(\text{Soc}(_RR)\)-semiregular. Then they study \(I\)-semiperfect modules. They prove that a ring \(R\) is \(I\)-semiperfect if and only if every simple factor module of \(_RR\) has projective \(I\)-cover. As a corollary they get equivalent conditions for a ring \(R\) to be \(Z(_RR)\)-semiperfect. Finally they prove that a ring \(R\) is \(Z_2(_RR)\)-semiperfect if and only if every simple factor module of \(_RR\) has a projective \(Z_2(_RR)\)-semicover.
scientific article; zbMATH DE number 5304605