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
.