Ring of Sets is Commutative Ring

Theorem
A ring of sets $\left({\mathcal R, *, \cap}\right)$ is a commutative ring whose zero is $\varnothing$.

Proof
By definition, the operations $\cap$ and $*$ are closed in $\mathcal R$.

Hence we can apply the following results:


 * Set System Closed under Symmetric Difference is Abelian Group: $\left({\mathcal R, *}\right)$ is an abelian group.


 * Set System Closed with Intersection is Commutative Semigroup: $\left({\mathcal R, \cap}\right)$ is a commutative semigroup.


 * Intersection Distributes over Symmetric Difference.


 * The identity of $\left({\mathcal R, *}\right)$ is $\varnothing$, and this, by definition, is the zero.

So $\left({\mathcal R, *, \cap}\right)$ is a commutative ring whose zero is $\varnothing$.