Positive_Elements_of_Ordered_Ring

Let $\struct {R, +, \circ, \le}$ be an ordered ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $P$ be the set of positive elements of $R$ , that is, $P = R_{\ge 0}$. Then $(1): \quad P + P \subseteq P$ $(2): \quad P \cap \paren {-P} = \set {0_R}$ $(3): \quad P \circ P \subseteq P$ If $\le$ is a total ordering, that is, if $\struct {R, +, \circ, \le}$ is a totally ordered ring, then $(4): \quad P \cup \paren {-P} = R$ The converse is also true Let $\struct {R, +, \circ}$ be a ring. Let $P \subseteq R$ such that $(1)$, $(2)$ and $(3)$ are satisfied. Then there is one and only one ordering $\le$ compatible with the ring structure of $R$ such that $P = R_{\ge 0}$. Also, if $(4)$ is also satisfied, then $\le$ is a total ordering.