In set-theory, one of the standard result (Bernstein's theorem) is that if there is an injection from $A$ to $B$ and an injection from $B$ to $A$, then there is a bijection from $A$ to $B$.
Consider similar situation on sets with some algebraic structures, say $A$ and $B$ are fields. If there is an injective homomorphism from $A$ into $B$ and also from $B$ into $A$, can we conclude that $A$ and $B$ are isomorphic fields?