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:
\(\ds x - y\) | \(\in\) | \(\ds L\) | ||||||||||||
\(\ds x \circ y\) | \(\in\) | \(\ds L\) | ||||||||||||
\(\ds x^{-1} \circ y\) | \(\in\) | \(\ds L\) |
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:
\(\ds x - y\) | \(\in\) | \(\ds T\) | ||||||||||||
\(\ds x \circ y\) | \(\in\) | \(\ds T\) | ||||||||||||
\(\ds x^{-1} \circ y\) | \(\in\) | \(\ds T\) |
So by definition of subset, $L \subseteq T$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.1$