Definition:Division Ring

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

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

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:


 * Every non-zero element of $$R$$is a unit.

This follows from is how a unit is defined.


 * $$R$$ has no proper elements.

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


 * $$R$$ has no zero divisors.

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