Example 3

A subring of an ordered ring, with the induced ordering, is an ordered ring.

In an ordered ring, the relations $x \geqslant y$ and $z \geqslant 0$ imply $xz \geqslant yz$. Indeed these inequalities are equivalent to $x - y \geqslant 0$, $z \geqslant 0$ and $(x - y)z \geqslant 0$ respectively. Analogously we can show that the relations $x \leqslant 0$ and $y \geqslant 0$ (resp. $y \leqslant 0$) imply $xy \leqslant 0$ (resp. $xy \geqslant 0$). These results are often invoked under the name of sign rules (two elements are said to have the same sign if they are both $\geqslant 0$ or both $\leqslant 0$). They imply that, if $\mathbf{A}$ is a totally ordered ring, then every square is positive, and in particular that every idempotent (for example the unit element) is positive.