Projective and free ordered modules

The paper introduces the concepts of $o$-free and $o$-projective modules over directed ring $R$. Some sufficient conditions are established under which all $o$-projective $R$-modules are $o$-free. In particular, it is proven that all $o$-projective $R$-modules are $o$-free in the cases: linearly ordered rings $R$ without divisors of zero in which each element $0<r<1$ is invertible; commutative factorable domain of integrity with any linear order; commutative rings without divisors of zero in which all projective modules are free with any linear order.