Chapter 3.3, Problem 11E

Define addition $\oplus$ and multiplication $\otimes$ on the set $\mathbb{Z}\times \mathbb{Z}$ as follows. For $a,b,c,d\in \mathbb{Z},\left(a,b\right)\oplus \left(c,d\right)=\left(a+c,b+d\right)$ and $\left(a,b\right)\otimes \left(c,d\right)=\left(ac,bd\right).$ Prove that $\left(\mathbb{Z}\times \mathbb{Z},\oplus ,\otimes \right)$ is a ring.