Definition:Division Ring/Definition 1

Definition
A division ring is a ring with unity $\struct {R, +, \circ}$ such that:
 * $\forall x \in R_{\ne 0_R}: \exists! x^{-1} \in R_{\ne 0_R}: x^{-1} \circ x = x \circ x^{-1} = 1_R$

where $R^*$ denotes the set of elements of $R$ without the ring zero $0_R$:
 * $R_{\ne 0_R} = R \setminus \set {0_R}$

That is, every non-zero element of $R$ has a (unique) non-zero product inverse.

Also see

 * Equivalence of Definitions of Division Ring