TITLE: Ideal generated by a irreducible element
QUESTION [7 upvotes]: Is the ideal generated by an irreducible element always a prime ideal in a ring?
If so why?

REPLY [3 votes]: HINT $\ $ Since $\rm\:(p)\:$ is prime iff $\rm\:p\:$ is prime, your assumption is equivalent to saying that all atoms (irreducibles) are prime. This hypothesis, combined with atomicity (every nonzero nonunit factors into a product of atoms) is equivalent to a domain being a UFD. So it is a very strong hypothesis - one that fails in any non-UFD number ring (which are always atomic by way of norms).