Positive, Nonnegative, Negative and Nonpositive Elements of an Ordered Ring

In the setting of Sets and Maps: Ordinary Notation, let $R$, with $+$, $\cdot$, $0$, $1$ and a total order $\le$ with strict relation $<$, be an ordered ring, and let $x\in R$.

¶ $x$ is positive if $0<x$, nonnegative if $0\le x$, negative if $x<0$, and nonpositive if $x\le0$.