Some Results on Regular Modules

In this paper we study the relationship between the projectivity and regularity. We show that projective module over a Von Neuman regular ring is regular. We also show that a projective module P is regular if every cyclic submodule of P is a direct summand of P and we shall see that if P is a projective regular module then J (P)=Z (P)=0 where J (P) and Z (P) are the Jacobson and singular submodules of P respectively. Also P is regular and indecomposable if P is projective ring with identity and all modules are unitary.