Definition:Ring of Sets

Definition
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.