Ring homomorphisms preserve structure

Ring homomorphisms preserve structure: Let $\varphi:R\to S$ be a ring homomorphism. Then for all $a,b\in R$,

If $\varphi$ is unital, then $\varphi(1_R)=1_S$. In particular, $\varphi(-a)=-\varphi(a)$ for all $a\in R$.

This is immediate from the definition of a ring homomorphism between rings ; when working with unital rings one typically requires $\varphi(1_R)=1_S$. These identities underpin the definitions of the kernel and image of $\varphi$.