# Definition:Division Ring

## Definition

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

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

where $R^*$ denotes the set of elements of $R$ without the ring zero $0_R$:

$R^* = R \setminus \left\{ {0_R}\right\}$

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

• Results about division rings can be found here.