Irreducible implies prime (PID)

From Commalg
Jump to: navigation, search

Statement

Verbal statement

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

Symbolic statement

Let R be a principal ideal domain and p an irreducible element in R. Then, if p|ab, then p|a or p|b.

Proof

Suppose p|ab and p \not | a. We need to prove that p|b.

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

1 = ax + py

for suitable choices of x,y \in R.

Multiplying both sides by b, we get:

b = abx + pby

Since p|ab, p divides the entire right-hand-side, and hence p|b.