Let $X\neq \emptyset$ be a set, and $\mathcal P(X)$ be the power set of $X$. Consider the operations $\Delta$ = symmetric difference (a.k.a. "XOR"), and $\bigcap$ = intersection. What is the principal ideal of $ T \subset X$?
I think it is the set of all subsets of $T$, $\mathcal P(T)$, since $T\Delta T=\emptyset$ and $( R\cap T) \Delta T $ is $T$ without $R$, for $R \subset X$. And also, $P(X) $ is a commutative ring so $P(X) \cap T \cap P(X)=P(X) \cap T=T \cap P(X)$