Intersection of Division Subrings is Division Subring

Theorem

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

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

Then the intersection $\ds \bigcap \mathbb K$ of the members of $\mathbb K$ is itself a division subring of $D$.

Proof

Let $\ds L = \bigcap \mathbb K$.

Let $0$ be the zero of $\struct {D, +, \circ}$.

By Intersection of Subgroups is Subgroup: General Result, $\struct {L, +}$ is a subgroup of $\struct {D, +}$.

By the One-Step Subgroup Test:

$\forall x, y \in L: x + \paren {-y} \in L$
$\struct {D \setminus \set 0, \circ}$ is a group
$\struct {K \setminus \set 0, \circ}$ is a group for each $K \in \mathbb K$

By Set Difference over Subset, $\struct {K \setminus \set 0, \circ}$ is a subgroup of $\struct {D \setminus \set 0, \circ}$ for each $K \in \mathbb K$.

$\ds L \setminus \set 0 = \bigcap_{K \mathop \in \mathbb K} \paren {K \setminus \set 0}$

By Intersection of Subgroups is Subgroup: General Result, $\struct {L \setminus \set 0, \circ}$ is a subgroup of $\struct {D \setminus \set 0, \circ}$.

 $\ds \forall x, y \in L: \,$ $\ds x \circ y$ $\in$ $\ds L$ $\ds \forall x \in L \setminus \set 0: \,$ $\ds x^{-1}$ $\in$ $\ds L \setminus \set 0$

By the Division Subring Test it follows that $\struct {L, +, \circ}$ is a division subring of $\struct {D, +, \circ}$.

$\blacksquare$