Intersection of Rings of Sets

Theorem
Let $\mathcal R_k$ be a ring of sets, where $k$ is an element of an arbitrary set of indices.

Then their intersection $\displaystyle \mathcal R = \bigcap_k \mathcal R_k$ is itself a ring of sets.

Proof
Consider the set $\displaystyle \mathcal S = \bigcup_k \mathcal R_k$.

Now let $S = \left\{{X \in Y: Y \in \mathcal S}\right\}$.

This contains all the elements of all the sets contained in all the $\mathcal R_k$.

Now consider the power set $\mathcal P \left({S}\right)$ of $S$.

By Power Set is Algebra of Sets and the definition of algebra of sets, we have that $\mathcal P \left({S}\right)$ is a ring of sets.

Thus $\left({\mathcal P \left({S}\right), *, \cap}\right)$ is a ring (in the abstract algebraic sense of the term).

From the method of construction of $\mathcal P \left({S}\right)$, it is clear that all the $\left({\mathcal R_k, *, \cap}\right)$ are subrings of $\mathcal P \left({S}\right)$.

The result then follows from Intersection of Subrings.