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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $\\le$ be a [[total order|0316AYH1SC]] on $R$.\nThe pair $(R, \\le)$ is an *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\nSo an ordered ring is a commutative [[partially ordered ring|A0NP6HQNPX]] whose order is total.\nExample: $\\Z$ with the usual order is an ordered ring, and so is every [[ordered field|010JCV9J4E]].\nExample: $\\Z[x]$ is an ordered ring if $p \u003e 0$ means that the leading coefficient of $p$ is positive.\nIn any ordered ring, $a^2 \\ge 0$ for all $a$: if $a \\ge 0$ this is clause 2, and if $a \u003c 0$ then $-a \u003e 0$ by clause 1 and $a^2 = (-a)^2$.\nNon-example: the ring $\\Z[i]$ of Gaussian integers has no such order.\nSquares would be nonnegative, so $1 = 1^2 \\ge 0$ and $-1 = i^2 \\ge 0$, and adding gives $0 \\ge 1$; with $1 \\ge 0$ this forces $1 = 0$.\n",
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