  Every homomorphism of modules is projective-stably equivalent to an
epimorphism but is not always to a monomorphism. We prove that a map is
projective-stably equivalent to a monomorphism if and only if its kernel is
torsionless, that is, a first syzygy. If it occurs although, there can be
various monomorphisms that are projective-stably equivalent to a given map. But
in this case there uniquely exists a "perfect" monomorphism to which a given
map is projective-stably equivalent.
