Definition:Strictly Positive

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

Then $x \in R$ is strictly positive iff $0_R \le x$ and $x \ne 0_R$.

One may also write (more conveniently) that $0_R < x$ or $x > 0_R$.

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


 * $R_{> 0_R} := \left\{{x \in R: x > 0_R}\right\}$

Alternative Notations
The notation $R_+^*$ is frequently seen for $R_{> 0_R}$, i.e. for $\left\{{x \in R: 0_R < x}\right\}$.

However, the notation $R_+$ and $R^+$ are also frequently seen for both $\left\{{x \in R: 0_R \le x}\right\}$ and $\left\{{x \in R: 0_R < x}\right\}$, and so suffer badly from ambiguity.

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

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

 * Strictly Negative


 * Positive