Symmetric Difference with Intersection forms Ring

Theorem
Let $$S$$ be a set.

Then $$\left({\mathcal{P} \left({S}\right), *, \cap}\right)$$ is a commutative ring with a unity which is $$S$$.

Proof

 * It has been established that $$\left({\mathcal{P} \left({S}\right), *}\right)$$ is an abelian group, where $$\varnothing$$ is the identity and each element is self-inverse.


 * From Power Set with Intersection is a Monoid, we know that $$\left({\mathcal{P} \left({S}\right), \cap}\right)$$ is a commutative monoid whose identity is $$S$$.


 * We have that Intersection Distributes over Symmetric Difference.


 * Thus $$\left({\mathcal{P} \left({S}\right), *, \cap}\right)$$ is a commutative ring with a unity which is $$S$$.


 * Next we find that $$\forall A \in \mathcal{P} \left({S}\right): A \cap \varnothing = \varnothing = \varnothing \cap A$$. Thus $$\varnothing$$ is indeed the zero.