# Irreducible implies prime (PID)

From Commalg

## Statement

### Verbal statement

In a principal ideal domain, any irreducible element is a prime element.

### Symbolic statement

Let be a principal ideal domain and an irreducible element in . Then, if , then or .

## Proof

Suppose and . We need to prove that .

Then, the ideal generated by and is principal, and is generated by a factor of both and . Since is irreducible, the only possibility for this is the whole ring. Thus:

for suitable choices of .

Multiplying both sides by , we get:

Since , divides the entire right-hand-side, and hence .