Ring Homomorphism

For two Rings $R$ and $S$, a mapping $f:R\to S$ is called a ring homomorphism if it satisfies the following conditions:

1. For any $(a,b)\in R$, $f(a+b)=f(a)+f(b)$.
2. For any $(a,b)\in R$, $f(ab)=f(a)\cdot f(b)$.
3. $f(1_R)=1_S$.

Here, the operations on the left-hand side of each condition are those defined in $R$, and the operations on the right-hand side are those defined in $S$. If such a mapping $f$ is bijective, it is called a Ring Isomorphism.