# 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 $\P \subset \mathbb S \subseteq \mathbb K$.

$\bigcap \mathbb S$ is the largest subring of $K$ contained in each of the elements of $\mathbb S$.
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.

$\blacksquare$