Intersection of Division Subrings Containing Subset is Smallest

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 $23.1$