Let M and N be free modules over Z . $M={(a,b)|a,b \in Z}, $ $N={ (2a,2b)|a,b \in Z } $. If M and N were
vector spaces, they would be isomorphic. However, because our scalar multiples are from
a ring that does not have multiplicative inverses, N has no vectors with odd-parity entries.
Thus, M and N are not isomorphic.