Definition:Negative/Ordered Ring

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Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$.

Then $x \in R$ is negative if and only if $x \le 0_R$.

The set of all negative elements of $R$ is denoted:

$R_{\le 0_R} := \set {x \in R: x \le 0_R}$

Also known as

The notations $R_-$ and $R^-$ are also frequently seen for $\set {x \in R: x \le 0_R}$.

However, these notations are also used to denote $\set {x \in R: x < 0_R}$, that is $R_{< 0_R}$, and hence are ambiguous.

Some treatments of this subject use the term define non-positive to define $x \in R$ where $0_R \le x$, reserving the term negative for what is defined on this website as strictly negative.

With the conveniently unambiguous notation that has been adopted on this site, the distinction between the terms loses its importance, as the symbology removes the confusion.

Also see