Prime Ideals of Ring of Integers

Theorem

Let $\struct {\Z, +, \times}$ denote the ring of integers.

Let $J$ be a prime ideal of $\Z$.

Then either:

$J = \set 0$

or:

$J = \ideal p$

where:

$p$ is a prime number

$\ideal p$ denotes the principal ideal of $\Z$ generated by $p$.

Proof

From Prime Ideal iff Quotient Ring is Integral Domain:

$J$ is a prime ideal of $\Z$ if and only if $\Z / J$ is an integral domain.

From Quotient Ring of Integers and Zero:

$\Z / \set 0 \cong \Z$

As $\Z$ is an integral domain, it follows that $\set 0$ is a prime ideal of $\Z$.

From Quotient Ring of Integers with Principal Ideal:

$\struct {\Z, +, \times} / \ideal p$ is isomorphic to $\struct {\Z_p, +_p, \times_p}$, the ring of integers modulo $p$.

From Ring of Integers Modulo Prime is Integral Domain:

$\struct {\Z_n, +_n, \times_p}$ is an integral domain if and only if $p$ is prime.

The result follows.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $13$