\begin{definition}[Definition:Division Ring/Definition 1]
A '''division ring''' is a ring with unity $\struct {R, +, \circ}$ such that:
:$\forall x \in R^*: \exists! x^{-1} \in R^*: x^{-1} \circ x = x \circ x^{-1} = 1_R$
where $R^*$ denotes the set of elements of $R$ without the ring zero $0_R$:
:$R^* = R \setminus \set {0_R}$
That is, every non-zero element of $R$ has a (unique) non-zero product inverse.
\end{definition}