Intersection of Subrings is Largest Subring Contained in all Subrings

Theorem
Let $\struct {R, +, \circ}$ be a ring.

Let $\mathbb L$ be a non-empty set of subrings of $R$.

Then the intersection $\displaystyle \bigcap \mathbb L$ of the members of $\mathbb L$ is the largest subring of $R$ contained in each member of $\mathbb L$.

Proof
Let $\displaystyle L = \bigcap \mathbb L$.

From Intersection of Subrings is Subring, $L$ is indeed a subring of $R$.

By Intersection of Subgroups is Subgroup, $\struct {L, +}$ is the largest subgroup of $\struct {R, +}$ contained in each member of $\mathbb L$.

By Intersection of Subsemigroups, $\struct {L, \circ}$ is the largest subsemigroup of $\struct {R, \circ}$ contained in each member of $\mathbb L$.

Let $x, y \in L$.

Then $\forall K \subseteq L$:
 * $x \circ y \in K \implies x \circ y \in L$
 * $x - y \in K \implies x - y \in L$

Thus, by the Subring Test, any $K \in \mathbb L: K \subseteq L$ and thus $L$ is the largest subring of $R$ contained in each member of $\mathbb L$.