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.