Partially ordered ring

Remarks

Example: $\R^n$ with componentwise addition and multiplication and the componentwise order is a partially ordered ring. Clause 1 holds coordinatewise, and the product of nonnegative real numbers is nonnegative, which gives clause 2. Non-example: $\R$ with the reversed order $a \preceq b \iff a \ge b$ satisfies clause 1 but not clause 2. Indeed $0 \preceq -1$, but $(-1)(-1) = 1$ and $0 \not\preceq 1$. The set $P = \set{a \in R : 0 \le a}$ is the positive cone. It satisfies $P + P \subseteq P$, $PP \subseteq P$ and $P \cap (-P) = \set{0}$, and the order is recovered as $a \le b \iff b - a \in P$.