Referring to the question:
Finitely generated ideals in a Boolean ring are principal, why?
How to prove: In every Boolean Ring
Does there exist any prime ideal in a Boolean Ring. Only Boolean ring I know is power set any set with symmetric difference and intersection. But there is no prime ideal as much as I can figure out.
Only case is $\mathbb Z_{2}$ also there are no prime ideals except $\{0\}$
Best Answer
Every ring with identity has a maximal ideal and every maximal ideal is prime. You can find several more proofs of these statements on the site, as well as online, or in any algebra book.
For concrete examples, just take $\prod \Bbb Z_2$, any number of copies of the field of two elements. You can produce a maximal ideal (many, actually) by picking a particular position and looking at the set of elements which are zero on that position.