Nonsingular CS-rings coincide with tight PP rings

It is shown that if M is a self-generator right R-module, then M is non-M-singular and CS iff M is M-tight and End(MR) is a right PP ring. In particular, right nonsingular right CS-rings R are precisely right PP and right R-tight. As applications we show, among others, that for any domain R, is right CS if and only if R is two-sided Ore domain and two-sided 2-hereditary, giving answer to an open question known previously in special cases. As another application, we show that for a von Neumann regular ring R, the matrix ring Mn(R), n>1, is right weakly selfinjective if and only if R is right selfinjective.