Quasi-projective modules over prime hereditary noetherian V-rings are projective or injective

Let Q be the field of rational numbers. As a module over the ring Z of integers, Q is Z-projective, but Q(Z) is not a projective module. Contrary to this situation, we show that over a prime right noetherian right hereditary right V-ring R, a right module P is projective if and only if P is R-projective. As a consequence of this we obtain the result stated in the title. Furthermore, we apply this to affirmatively answer a question that was left open in a recent work of Holston, Lopez-Permouth and Orhan Ertag (2012) [9] by showing that over a right noetherian prime right SI-ring, quasi-projective right modules are projective or semisimple. (c) 2012 Elsevier Inc. All rights reserved.