I have the following definitions:
An ideal $I$ is prime if, whenever $ab \in I$, either $a \in I$ or $b \in I$.
An ideal $J$ is irreducible if, whenever $J = I_1 \cap I_2$ for ideals $I_1 $ and $ I_2$, $J \subseteq I_1$ or $J \subseteq I_2$. EDIT: As per Arturo Magidin's correction, we must have $ J = I_1$ or $J = I_2$.
Under what conditions are prime ideals irreducible ideals? What about the converse?
I'd guess that:
i) prime ideals are always irreducible (mirroring the result for elements of a ring) [provided we're in an integral domain]
ii) irreducible ideals aren't always prime. Are there any nice examples?
iii) In certain types of ring, irreducible ideals are always prime (e.g. a UFD or stronger). I'd guess this mirrors the result for elements
iv) The ideal $ \langle a\rangle$ is prime iff $a$ is prime, and similarly in the irreducible case. So in PIDs, prime ideals are irreducible ideals and they are generated by a (prime) irreducible.
Am I correct? How would I go about proving these (especially iii)?