Originally Posted by

**matt.qmar** Hi... I am a little unsure of this result...

I think I have found a prime ideal which is not maximal.

$\displaystyle \{0\} \oplus \mathbb{Z}$ is a prime ideal of $\displaystyle \mathbb{Z} \oplus \mathbb{Z}$

but

$\displaystyle \{0\} \oplus \mathbb{Z}$ is (properly) contained in $\displaystyle 2\mathbb{Z} \oplus \mathbb{Z}$, also an ideal of $\displaystyle \mathbb{Z} \oplus \mathbb{Z}$, so $\displaystyle \{0\} \oplus \mathbb{Z}$ is not maximal...

so I think there is an issue here?