# In a Euclidean ring, could we prove irreducible implies prime directly?

In a UFD, primeness and irreducibility are equivalent. In particular, every Euclidean ring which is an integral domain is a UFD.

My question is this: is it possible to prove that "irreducible implies prime" directly (I mean without invoking "Euclidean implies PID implies UFD").

• If you can get to the existence of gcd, it can be finished from there. However from gcd I think is only a small step to PID anyway. – coffeemath Nov 8 '14 at 3:51