# Ring of Sets is Commutative Ring

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## 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 under Intersection is Commutative Semigroup: $\left({\mathcal R, \cap}\right)$ is a commutative semigroup.

- 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 $\varnothing$.

$\blacksquare$