Symmetric Difference with Intersection forms Ring/Proof 1

Proof
From Symmetric Difference on Power Set forms Abelian Group, $\struct {\powerset S, \symdif}$ is an abelian group, where $\O$ is the identity and each element is self-inverse.

From Power Set with Intersection is Monoid, $\struct {\powerset S, \cap}$ is a commutative monoid whose identity is $S$.

Also Intersection Distributes over Symmetric Difference.

Thus $\struct {\powerset S, \cap}$ is a commutative ring with a unity which is $S$.

From Intersection with Empty Set:
 * $\forall A \in \powerset S: A \cap \O = \O = \O \cap A$

Thus $\O$ is indeed the zero.

However, from Set Intersection Not Cancellable, it follows that $\struct {\powerset S, \symdif, \cap}$ is not an integral domain.