Intersection of Division Subrings is Largest Division Subring Contained in all Division Subrings

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {D, +, \circ}$ be a division ring.

Let $\mathbb K$ be a non-empty set of division subrings of $D$.

Let $\bigcap \mathbb K$ be the intersection of the elements of $\mathbb K$.


Then $\bigcap \mathbb K$ is the largest division subring of $D$ contained in each element of $\mathbb K$.


Proof

Let $L = \bigcap \mathbb K$.

From Intersection of Division Subrings is Division Subring, $\struct {L, +, \circ}$ is a division subring of $\struct {D, +, \circ}$.


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

By Intersection of Subgroups is Subgroup, $\struct {L, \circ}$ is the largest subgroup of $\struct {D, \circ}$ contained in each element of $\mathbb K$.


Let $S$ be a subring of $D$ such that:

$\forall K \in \mathbb K: S \subseteq K$

By Intersection is Largest Subset, $S \subseteq L$.

Thus $L$ is the largest division subring of $D$ contained in each element of $\mathbb L$.

$\blacksquare$


Sources