Definition:Strictly Negative

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

Then $x \in R$ is strictly negative :
 * $x \le 0_R$ and $x \ne 0_R$

That $x$ is strictly negative may be (more conveniently) denoted $0_R < x$ or $x > 0_R$.

Thus, the set of all strictly negative elements of $R$ is denoted:


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

Also known as
The notation $R_-^*$ is frequently seen for $R_{< 0_R}$, that is for $\set {x \in R: 0_R > x}$.

However, the notations $R_-$ and $R^-$ are also frequently seen for both $\set {x \in R: 0_R \ge x}$ and $\set {x \in R: 0_R > x}$, and so suffer badly from ambiguity.

Some treatments of this subject reserve the term negative to define $x \in R$ where $0_R > x$, using the term non-positive for what is defined on this website as 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

 * Definition:Negative
 * Definition:Positive
 * Definition:Strictly Positive