Let $\phi:R\rightarrow R'$ be a (not necessarily unital) ring homomorphism. If $S'$ is a subring of $R'$, it is easy to show that $\phi^{-1}(S')$ is a subring of $R$. But if $S'$ is also unital, will $\phi^{-1}(S')$ be unital as well? In other words, can we always find a multiplicative identity in $\phi^{-1}(S')$?
PS: By a subring $S$ of $R$ I mean a subset $S\subseteq R$ such that it is a ring itself under the same operations of $R$.