let $R$ and $S$ be unitary rings and $\phi:R\rightarrow S$ a ring homomorphism. is the following correct:
$\phi(1_R\cdot1_R)=\phi(1_R)\cdot\phi(1_R)$ so $\phi(1_R)(1_S-\phi(1_R))=0_S$ and so $\phi(1_R)$ could be anything in $S$ when $S$ is a general ring, i mean we can not conclude what values $\phi(1_R)$ could take in $S$ . But when $S$ is an integral domain then we can say that we have $\phi(1_R)=0_S$ or $1_S-\phi(1_R)=0_S$ i.e, $\phi(1_R)=1_S$. Moreover when $\phi$ is a monomorphism then since only $0_R$ maps to $0_S$ then necessarily $\phi(1_R)=1_S$.