Projective dimension in UFDs

A characterization of factorial Noetherian domains in terms of 0-submodules is presented and it is applied to give a criterion for the ring of a hypersurface to be factorial (Vasconcelos, Computational Methods in Commutative Algebra and Algebraic Geometry, Springer, Berlin, 1998) and to reduce a conjecture of Evans and Grffith in Syzygies (Cambridge University Press, Cambridge, UK, 1985) to ideals.