Intersection of Division Subrings Containing Subset is Smallest

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Theorem

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

Let $S \subseteq D$ be a subset of $D$.

Let $L$ be the intersection of the set of all division subrings of $D$ containing $S$.


Then $L$ is the smallest division subring of $D$ containing $S$.


Proof

From Intersection of Division Subrings is Division Subring, $L$ is indeed a division subring of $D$.

Let $T$ be a division subring of $D$ containing $S$.

Let $x, y \in L$.

By the Division Subring Test, we have that:

\(\displaystyle x - y\) \(\in\) \(\displaystyle L\)
\(\displaystyle x \circ y\) \(\in\) \(\displaystyle L\)
\(\displaystyle x^{-1} \circ y\) \(\in\) \(\displaystyle L\)

By Intersection is Largest Subset, it follows that $x, y \in T$.


But $T$ is also a division subring of $D$.

So, by the Division Subring Test again, we have that:

\(\displaystyle x - y\) \(\in\) \(\displaystyle T\)
\(\displaystyle x \circ y\) \(\in\) \(\displaystyle T\)
\(\displaystyle x^{-1} \circ y\) \(\in\) \(\displaystyle T\)

So by definition of subset, $L \subseteq T$.

$\blacksquare$


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