In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let ${pi}$ be a group and let M ${ ightarrow}$ N be a homomorphism between projective $mathbb{Z}[{pi}]$-modules such that $mathbb{Z}_p;{otimes}_{mathbb{Z}[{pi}]}M{ ightarrow}mathbb{Z}_p{otimes}_{mathbb{Z}[{pi}]};N$ is injective; for which other right $mathbb{Z}[{pi}]$-modules V is the induced map $V{otimes}_{mathbb{Z}[{pi}]};M{ ightarrow};V{otimes}_{mathbb{Z}[{pi}]};N$ also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.