# Definition:Negative/Ordered Ring

## Definition

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.