Definition:Division Ring

Definition
A division ring is a ring with unity $\left({R, +, \circ}\right)$ such that:


 * $\forall x \in R_{\neq0}: \exists ! x^{-1} \in R^*: x^{-1} \circ x = x \circ x^{-1} = 1_R$

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

Alternative definitions
A division ring is a ring with unity $\left({R, +, \circ}\right)$ such that:


 * $(1): \quad$ Every non-zero element of $R$ is a unit.

This follows from how a unit is defined.


 * $(2): \quad R$ has no proper elements.

This follows from the fact that a unit is not a proper element.


 * $(3): \quad R$ has no proper zero divisors.

This follows from the fact that a unit can not be a zero divisor.

Also see

 * Definition:Field (Abstract Algebra)
 * Definition:Skew Field