Ring of Sets is Commutative Ring

Theorem
A ring of sets $\struct {\RR, *, \cap}$ is a commutative ring whose zero is $\O$.

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

Hence we can apply the following results:


 * Set System Closed under Symmetric Difference is Abelian Group: $\struct {\RR, *}$ is an abelian group.


 * Set System Closed under Intersection is Commutative Semigroup: $\struct {\RR, \cap}$ is a commutative semigroup.


 * Intersection Distributes over Symmetric Difference.


 * The identity of $\struct {\RR, *}$ is $\O$, and this, by definition, is the zero.

So $\struct {\RR, *, \cap}$ is a commutative ring whose zero is $\O$.