Free $f$-modules

Let $R$ be a directed ring whose cone of positive elements is strict and satisfies Ore's condition. The main result: there exists a freef-module over an $o$-module $_RM$ with cone $P_M$ if and only if $P_M$ is half-closed (this generalizes Vainberg's theorem for ordered Abelian groups). In this connection various characterizations off-modules with strict cones are given.