Intersection of All Ring Ideals Containing Subset is Smallest

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

Let $S \subseteq R$ be a subset of $R$.

Let $L$ be the intersection of the set of all ideals of $R$ containing $S$.

Then $L$ is the smallest ideal of $R$ containing $S$.

Proof
From Intersection of Subrings Containing Subset is Smallest, $L$ is the smallest subring of $R$ containing $S$.

From Intersection of Ring Ideals is Ideal‎, $L$ is an ideal of $R$.

As $L$ is the smallest subring of $R$ contained in each member of $\mathbb L$, and it is an ideal of $R$, there can be no smaller ideal as it would then not be a subring.

So the intersection of the set of all subrings of $R$ containing $S$ is the smallest subring of $R$ containing $S$.