In the integral domain every prime is irreducible. But the converse is not true, for example, $1+\sqrt{-3}$ is an irreducible but not a prime in ${\Bbb Z}[\sqrt{-3}]$. In a UFD, "prime" and "irreducible" are equivalent.
Here is my question:
Is there a non-UFD integral domain such that prime is equivalent to irreducible?
Best Answer
Yes, e.g. its vacuously true in the ring of all algebraic integers, which has no irreducibles (so no primes), since $\rm\: a = \sqrt{a} \sqrt{a}.\:$ Such domains are known as antimatter domains, since they have no atoms (irreducibles).