Definition:Division Ring

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Division

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.

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.