division ring

Definition

A ring $R$ is a division ring if every non-zero element $x\in R\backslash 0$ is a unit. That is, there exists $y,z\in R$ such that

$$xy=1\text{and}zx=1$$

Examples

The quaternions $\mathbb{H}$ is a division ring that is not a field.

The polynomial ring $R[x]$ is never a division ring as $x\in R[x]$ is not a unit.