4. Facts valid in distributive lattices

(a) $I \vee J = \{i \vee j:i \in I, j \in J\}$. (Contrast with §(c) above.)

(b) Ideals$(L)$ is also a distributive lattice.

(c) Any meet-irreducible element of Ideals$(L)$ is a prime ideal.

(d) Theorem. Every ideal $I _ 0$ is the intersection of those prime ideals that contain $I _ 0$.

Equivalently,

(d$'$) For an ideal $I _ 0$ and $a \not \in I _ 0$, there is a prime ideal $I$ with $I _ 0 \subseteq I$ but $a \not \in I$.

(e) Corollary. In a distributive lattice, prime ideals separate points. (In other words, given any two distinct elements, there is a prime ideal that contains one and not the other.)

(f) Any maximal ideal is prime.