$\varphi:R\rightarrow S$ is said to be a ring homomorphism if, $R,S$ are rings and $\varphi$ is a map such that:
In the definition of a ring isomorphism, the third condition isn't stated, but here, we need the third condition. My question is, why do we need to state explicitly that the identity in $R$ is mapped to the identity in $S$. Can't we just deduce it from the definitions of a ring homomorphism?
EDIT: In reference to Anurag A's comment that rings are sometimes defined without the existence of unity, all the rings I am looking at will have unity in them, so if the answers are restricted to the case for when they do have unity, that would be extremely helpful