Set of Subrings forms Complete Lattice

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

Let $\mathbb K$ be the set of all subrings of $K$.

Then $\struct {\mathbb K, \subseteq}$ is a complete lattice.

Proof
Let $\O \subset \mathbb S \subseteq \mathbb K$.

By Intersection of Subrings is Largest Subring Contained in all Subrings:
 * $\bigcap \mathbb S$ is the largest subring of $K$ contained in each of the elements of $\mathbb S$.

By Intersection of Subrings Containing Subset is Smallest:
 * The intersection of the set of all subrings of $K$ containing $\bigcup \mathbb S$ is the smallest subring of $K$ containing $\bigcup \mathbb S$.

Thus:
 * Not only is $\bigcap \mathbb S$ a lower bound of $\mathbb S$, but also the largest, and therefore an infimum.


 * The supremum of $\mathbb S$ is the intersection of the set of all subrings of $K$.

Therefore $\struct {\mathbb K, \subseteq}$ is a complete lattice.