Ordered ring

Remarks

So an ordered ring is a commutative partially ordered ring whose order is total. Example: $\Z$ with the usual order is an ordered ring, and so is every ordered field. Example: $\Z[x]$ is an ordered ring if $p > 0$ means that the leading coefficient of $p$ is positive. In any ordered ring, $a^2 \ge 0$ for all $a$: if $a \ge 0$ this is clause 2, and if $a < 0$ then $-a > 0$ by clause 1 and $a^2 = (-a)^2$. Non-example: the ring $\Z[i]$ of Gaussian integers has no such order. Squares would be nonnegative, so $1 = 1^2 \ge 0$ and $-1 = i^2 \ge 0$, and adding gives $0 \ge 1$; with $1 \ge 0$ this forces $1 = 0$.