Division Subring Test

Theorem
Let $\left({K, +, \circ}\right)$ be a division ring, and let $L$ be a subset of $K$.

Then $\left({L, +, \circ}\right)$ is a division subring of $\left({K, +, \circ}\right)$ iff these all hold:


 * $(1) \quad L^* \ne \varnothing$


 * $(2) \quad \forall x, y \in L: x + \left({-y}\right) \in L$


 * $(3) \quad \forall x, y \in L: x \circ y \in L$


 * $(4) \quad x \in L^* \implies x^{-1} \in L^*$

Necessary Condition
Suppose $\left({L, +, \circ}\right)$ is a division subring of $\left({K, +, \circ}\right)$.

The conditions $(1)$ to $(3)$ hold by virtue of the Subring Test.

Then $(4)$ also holds by the definition of a division ring:
 * $\forall x \in L^*: \exists ! x^{-1} \in L^*: x^{-1} \circ x = x \circ x^{-1} = 1_L$

Sufficient Condition
Suppose the conditions $(1)$ to $(4)$ hold.

By $(1)$ to $(3)$, it follows from Subring Test that $\left({L, +, \circ}\right)$ is a subring of $\left({K, +, \circ}\right)$.

By $(4)$, every element of $L^*$ has a product inverse.

Thus, from the Two-Step Subgroup Test, $\left({L^*, \circ}\right)$ is a group.

Therefore, $\left({L, +, \circ}\right)$ is a ring in which every element has a product inverse, which makes $\left({L, +, \circ}\right)$ a division ring.