Definition of a Ring Homomorphism

Let $R$ and $S$ be rings. A ring homomorphism from $R$ to $S$ is a mapping $\theta :R\to S$ that satisfies:

• $\theta (a+b)=\theta (a)+\theta (b)$, and
• $\theta (ab)=\theta (a)\theta (b)$, for all $a,b\in R$.