show set is prime ideal

Question

Let I = { (a,0): a E Z}

A)show that I is a prime ideal of Z X Z

B) by considering (ZXZ)/I , or otherwise , determine whether I is a maximal ideal of ZXZ.

(0,0) is in I so I is non-empty

let (a,0) , (b,0) E I

than

(a,0)-(b,0) = (a-b,0) which is in I

for any (m,n) in ZXZ

(m,n)(a,0) = (am,0) which is in I this I is an ideal.

How do I show it is prime?

also need help with part b, not sure how to start?

thank you.

Answer 1

$(\mathbb{Z} \times \mathbb{Z}) /I \cong \mathbb{Z}$, by the map $f(x,y)=y$. So the quotient ring is an integral domain, hence $I$ is prime. But since $\mathbb{Z}$ is not a field, then $I$ is not maximal.