Let X be a set then $\displaystyle (\matchal{P}(X),\Delta,\cap)$ forms a commutative ring with unity X. Although not a principal ideal domain if X has at least two members, I was wondering if it was necessarily a principal ideal ring.

Namely if I was an ideal whether $\displaystyle I=(\bigcup I)$ necessarily.

I think it's obvious if X is a finite set, but for infinite X I can't decide as it's closed only under finite symmetric differences.

This is just for personal curiosity.