Set of Subrings forms Complete Lattice

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


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.

$\blacksquare$


Sources