Ring Theory, Prime Number

Question

Consider the ring of integers $\mathbb Z$. Prove that if $p$ is a prime number, then $\langle p \rangle$ is a maximal ideal.

I know I need to show that there is no ideal $J$ such that $\langle p \rangle$ is a subset of $J$. But I do not know how to do it.

Answer 1

Exercise: in a ring with $1$, if an ideal contains $1$, then it is the whole ring.

First prove the above statement; now assume that $J$ contains $p\mathbb{Z}$ and some element that is not a multiple of $p$, i.e. is strictly bigger than $p\mathbb{Z}$, and prove that $J$ must be everything.

Sorry, there really is no point in breaking this exercise down any further.

Answer 2

Hints

1. Prove that $\langle a\rangle\subseteq\langle b\rangle$ if and only if $b\mid a$ ($b$ divides $a$).
2. Every ideal in $\mathbb{Z}$ is principal.