Modules over bounded Dedekind prime rings (Q5890480)

The central result of this paper shows that the concepts of dividing module and \(\pi\)-projective module coincide for modules over a bounded Dedekind prime ring, and gives a complete description of such modules.NEWLINENEWLINENEWLINELet \(R\) be a ring and let \(M\) be an \(R\)-module. Then \(M\) is said to be a dividing module if \(\Hom(M,X+Y)=\Hom(M,X)+\Hom(M,Y)\) for all submodules \(X\) and \(Y\) of \(M\) (where the Hom-sets are considered to be subsets of \(\Hom(M,M)\)). The module \(M\) is said to be \(\pi\)-projective if for all submodules \(U\) and \(V\) of \(M\) with \(U+V=M\) there is an endomorphism \(f\) on \(M\) such that \(f(M)\subseteq U\) and \((1-f)(M)\subseteq V\). The class of \(\pi\)-projective modules includes all projective modules and all dividing modules. It is shown that if \(R\) is a bounded Dedekind prime ring then an \(R\)-module \(M\) is dividing if and only if it is \(\pi\)-projective, and that \(M\) is dividing if and only if it satisfies one of four conditions; one condition is that \(M\) is projective; the other three conditions give very precise structural descriptions of \(M\) in the non-projective case according as \(M\) is torsion, mixed, or torsion-free (and in this case \(R\) itself must also have a precise matrix structure). For instance, in the mixed case \(M=X\oplus Y\) where \(X\) is a non-zero torsion injective module with each of its primary components indecomposable, and \(Y\) is a non-zero finitely-generated projective module.