Irreducible ideal

Remarks

Example: every prime ideal $P$ is irreducible; if $P = J \cap K$ with $a \in J \setminus P$ and $b \in K \setminus P$, then $ab \in J \cap K = P$, a contradiction. Non-example: $(6)$ in $\Z$ fails clause 2, since $(6) = (2) \cap (3)$ with both terms strictly larger. In a Noetherian ring, every irreducible ideal is primary. The converse fails: in $k[x, y]$ over a field $k$, the ideal $(x^2, xy, y^2)$ is primary, but it equals $(x^2, y) \cap (x, y^2)$. Some authors call an ideal with clause 2 alone meet-irreducible, and then allow $I = R$.