Proving that primes are irreducible in any integral domain is simple.
Assume that a prime is not irreducible. Then
, where
and
are not units. Now we know that
. Therefore,
. Let
. Then
, where
. Now we have
. This implies
, which is a contradiction as
cannot be a unit.
But proving that every irreducible is a prime too requires more specialized conditions. Say is an irreducible such that
. Also assume that
. If euclidean division is valid in the ring, then
. If every ideal has to be a PID, then
.
This might not be an exhaustive list of conditions for which if
and
are co-prime, but this is all we have right now.
Anyway, if and
, then
. Multiplying by
on both sides, we ultimately get
, which proves
. This is valid for both PIDs and Euclidean domains. Note that the commutativity of the domain makes all this possible.