Non-Zero Elements of Division Ring form Group

Theorem
If $$\left({R, +, \circ}\right)$$ is a division ring, then $$\left({R^*, \circ}\right)$$ is a group.

Proof

 * A division ring by definition is a ring with unity, and therefore not null.


 * A division ring by definition has no zero divisors, so $$\left({R^*, \circ}\right)$$ is a semigroup.


 * $$1_R \in \left({R^*, \circ}\right)$$ and so the identity of $$\circ$$is in $$\left({R^*, \circ}\right)$$.


 * By the definition of a division ring, each element of $$\left({R^*, \circ}\right)$$ is a unit, and therefore has a unique inverse in $$\left({R^*, \circ}\right)$$.

Thus $$\left({R^*, \circ}\right)$$ is a semigroup with an identity and inverses and so is a group.