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  "body": "\n## Definition\n\nA *lattice-ordered ring* is a [[partially ordered ring|A0NP6HQNPX]] $(R, \\le)$ whose order is a [[lattice|WH4SFAACAS]], with meet $\\wedge$ and join $\\vee$.\nA lattice-ordered ring $R$ is an *f-ring* if for all $a, b, c \\in R$,\n$$a \\wedge b = 0 \\text{ and } c \\ge 0 \\implies ca \\wedge b = 0 \\text{ and } ac \\wedge b = 0.$$\n\n## Remarks\n\nExample: the ring $C(X)$ of [[continuous real functions|0P27W9RW0G]] on a topological space $X$, with the pointwise order, is an f-ring.\nIf $\\min(a, b) = 0$ pointwise and $c \\ge 0$, then at each point where $b \u003e 0$ one has $a = 0$, so $ca = 0$ there, and $\\min(ca, b) = 0$.\nNon-example: the ring of real $2 \\times 2$ matrices with the entrywise order is lattice-ordered but is not an f-ring.\nLet $E_{ij}$ be the matrix unit; then $E_{11} \\wedge E_{21} = 0$ and $E_{21} \\ge 0$, but $E_{21} E_{11} = E_{21}$ and $E_{21} \\wedge E_{21} = E_{21} \\ne 0$.\nEvery totally ordered ring is an f-ring, and in an f-ring every square is nonnegative.\n",
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