Definition 1

— Given a commutative ring $\mathbf{A}$, we say that an ordering on $\mathbf{A}$ is compatible with the ring structure of $\mathbf{A}$ if it is compatible with the additive group structure of $\mathbf{A}$, and if it satisfies the following axiom : (OR) The relations $x \geqslant 0$ and $y \geqslant 0$ imply $xy \geqslant 0$. The ring $\mathbf{A}$, together with such an ordering, is called an ordered ring.