Definition:Ring of Sets

A ring of sets $$\mathcal {R}$$ is a non-empty system of sets such that for all $$A, B \in \mathcal {R}$$: where $$\cap$$ denotes set intersection and $$*$$ denotes set symmetric difference.
 * $$A \cap B \in \mathcal{R}$$;
 * $$A * B \in \mathcal{R}$$

That is, the operations $$\cap$$ and $$*$$ are closed in $$\mathcal {R}$$.

A ring of sets when considered as an algebraic structure $$\left({\mathcal{R}, *, \cap}\right)$$ is a commutative ring.

As shown here, a ring of sets is also a semiring of sets.