Morphisms represented by monomorphisms

We answer a question posed by Auslander and Bridger. 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, 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.