Hello, I am stuck with these questions

1. Show that $\displaystyle \mathbb{Z} \oplus 2\mathbb{Z}$ is a maximal ideal in $\displaystyle \mathbb{Z} \oplus \mathbb{Z}$

2. Let $\displaystyle R$ be a commutative ring with unity. Prove that every prime ideal of $\displaystyle R$ is also a maximal ideal of $\displaystyle R$.

Thanks in advance for your help.