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 write (more conveniently) that $0_R < x$ or $x > 0_R$ to express that $x$ is strictly positive.

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

Sources

• Seth Warner: Modern Algebra (1965): $\S 1$
• Seth Warner: Modern Algebra (1965): $\S 23$