Projective and free ordered modules

The paper introduces the concepts ofo-free ando-projective modules over directed ring R. Some sufficient conditions are established under which allo-projective R-modules areo-free. In particular, it is proven that allo-projective R-modules areo-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.Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 41–52, January, 1972.