A left $R$-module $M$ is flat if and only if every morphism $f:K\to M$ , where $K$ finitely presented, factors through a finitely generated projective left $R$-module.
I have found a proof of this theorem in T.Y.Lam's book "Lectures on Modules and Rings", they use the Equational Criteria for flatness. But I am looking for other proofs without using linear equations. Can any one help me?