Quasi-projective modules over prime hereditary Noetherian V-rings are projective or injective. (Q1758180)

The question that is investigated in this paper, is for which Noetherian hereditary domains \(D\), \(D\)-projectivity of \(D\)-modules will ensure projectivity. This question has its origin in the fact that unlike the case of injectivity, projectivity is not implied by \(R\)-projectivity. As an example of this, it is shown that the ring of rational numbers \(\mathbb Q\), as a module over the integers \(\mathbb Z\), is \(\mathbb Z\)-projective, but not projective. The main result states that \(D\)-projectivity and projectivity are equivalent for a right module over a right Noetherian right hereditary right \(V\)-domain \(D\). This result is proved via two lemmas that give results on \(R\)-projectives and projectives over right nonsingular right \(V\)-rings and right hereditary domains, respectively. As a corollary, the main result is extended to prime right Noetherian right hereditary right \(V\)-rings.