Difference between maximal and maximum ideal.

Question

For example, I consider this: $\ p\mathbb{Z}$ is maximal ideal of $\mathbb{Z}$ iff $\ p$ is prime. But my question is for two distinct primes $ p_1$ and $ p_2$, which one will be maximal ideal (namely $ p_1\Bbb{Z}$ and $ p_2\Bbb{Z}$)?

$\ M\subset R$ is a maximal ideal if $\exists$ an ideal $\ U$ of $\ R$ such that $\ M \subset U \subset R$ implies either $\ M=U$ or $\ U=R$.

So i) how can two distinct ideals $\ p_1\mathbb{Z}$ and$\ p_2\mathbb{Z}$ both be maximal?

ii) Or can I say a maximal ideal is actually a maximum ideal?

(As I know $ p_1\Bbb{Z} \not\subset p_2\Bbb{Z}$ and vice-versa)

Answer 1

Both ideals are maximal. Rings can have multiple maximal ideals, and usually do so.

In fact, rings which have only a single maximal ideal are called local rings and are "comparatively simple".

Answer 2

Both are maximal. Indeed,

$$\mathbb{Z}/2\mathbb{Z},\quad \mathbb{Z}/3\mathbb{Z}$$

are fields.

There is no contradiction because neither of the inclusions $2 \mathbb{Z} \subseteq 3 \mathbb{Z}$ and $3\mathbb{Z}\subseteq 2 \mathbb{Z}$ hold.