Let Q be the field of rational numbers. As a module over the ring Z of integers, Q is Z-projective, but QZ 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, López-Permouth and Orhan Ertag (2012) by showing that over a right noetherian prime right SI-ring, quasi-projective right modules are projective or semisimple.
Dinh, Hai Q; Holston, Christopher J; Huynh, Dinh V (2012). Quasi-Projective Modules Over Prime Hereditary Noetherian V-rings are Projective or Injective. Elsevier 360 87-91. doi: 10.1016/J.JALGEBRA.2012.04.002. Retrieved from https://oaks.kent.edu/mathpubs/12