Note on almost relative projectives and almost relative injectives (Q1802980)

The first part of the paper deals with relative projective modules ((almost) \(M\)-projective, almost \(M\)-simple-projective) and characterizes such modules over perfect (semi-perfect) rings. The second part is then devoted to the dual, injective, notions and their properties. The last two parts investigate the condition (D) ``Any submodule \(N\) of a direct sum \(M\) of its submodules \(M_ i\) such that the projection of \(N\) to some \(M_ i\) is onto, contains a non-zero direct summand of \(M\)'' and its dual.