Question on the power set as a ring and its ideals

Hey guys I've a question for you. I was in the middle of studying a bit the power set of any set as a boolean ring with the symmetric difference and the intersection as operations without using any resource or help as an exercise. I was thinking of what could these rings ideals be and I came to the conclusion that the ideal generated by an element of P(X) must be the power set of that element. I also came to the conclusion that P(X) is a local ring and X is the only invertible element.

I was trying to now verify if the ring is noetherian or a p.i.d. but I'm kind of stuck on that.
I can kinda easily see it be non-noetherian (if X is infinite) as I can take the chain <An> with A0 in P(X), A(n+1) = An U {a} with a not in An. Essentialy I've built an infinite strictly ascending chain of ideals.

On the other hand I can see that any ideal generated by two elements is actually equal to the ideal generated by the union of those elements. This can easily be extended for any finite number of generators. So my question is: am I missing something or is it possible that every finitely generated ideal is principal BUT the ring is non-noetherian?