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 with Symmetric Difference is Group: $$\left({\mathcal{R}, *}\right)$$ is an abelian group.


 * Set System Closed with Intersection is 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 $$\varnothing$$.