# Definition:Positivity Property

## Definition

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

Let $P \subseteq R$ such that:

\((\text P 1)\) | $:$ | \(\ds P + P \subseteq P \) | |||||||

\((\text P 2)\) | $:$ | \(\ds P \cap \paren {-P} = \set {0_R} \) | |||||||

\((\text P 3)\) | $:$ | \(\ds P \circ P \subseteq P \) |

Let $\PP: R \to \set {\T, \F}$ be the propositional function defined as:

- $\forall x \in D: \map \PP x \iff x \in P$

Then $\PP$ is the **positivity property** on $\struct {R, +, \circ}$.

## Also defined as

The name **positivity property** is also defined to be the similar propositional function, usually defined on an integral domain $\struct {D, +, \times}$ which does not include zero in its fiber of truth.

Because $\struct {R, +, \circ}$ may have (proper) zero divisors, such a propositional function may not be closed under $\circ$.

Hence it is the intention on $\mathsf{Pr} \infty \mathsf{fWiki}$ to refer consistently to that propositional function as the **strict positivity property**, and to reserve **positivity property** for this one.

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers: Theorem $23.12$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 6$: Rings and fields