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.
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