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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $\\le$ be a [[partial order|011C29Q1S1]] on $R$.\nThe pair $(R, \\le)$ is a *partially ordered ring* if for all $a, b, c \\in R$:\n\n1. if $a \\le b$, then $a + c \\le b + c$;\n2. if $0 \\le a$ and $0 \\le b$, then $0 \\le ab$.\n\n## Remarks\n\nExample: $\\R^n$ with componentwise addition and multiplication and the componentwise order is a partially ordered ring.\nClause 1 holds coordinatewise, and the product of nonnegative real numbers is nonnegative, which gives clause 2.\nNon-example: $\\R$ with the reversed order $a \\preceq b \\iff a \\ge b$ satisfies clause 1 but not clause 2.\nIndeed $0 \\preceq -1$, but $(-1)(-1) = 1$ and $0 \\not\\preceq 1$.\nThe set $P = \\set{a \\in R : 0 \\le a}$ is the *positive cone*.\nIt satisfies $P + P \\subseteq P$, $PP \\subseteq P$ and $P \\cap (-P) = \\set{0}$, and the order is recovered as $a \\le b \\iff b - a \\in P$.\n",
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