Non-Zero Elements of Division Ring form Group

Theorem

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

Then $\struct {R^*, \circ}$ 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 $\struct {R^*, \circ}$ is a semigroup.

$1_R \in \struct {R^*, \circ}$ and so the identity of $\circ$ is in $\struct {R^*, \circ}$.

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

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

$\blacksquare$