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  "body": "\n## Definition\n\nLet $R$ be an associative [[ring|0B1261B5ED]] with identity, and let $U$ be its [[group of units|07C6ECNJET]].\nA pair $(a, b) \\in R^2$ is *admissible* if there are $c, d \\in R$ such that the matrix $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$ lies in the [[general linear group|0FCHE902D7]] $\\mathrm{GL}(2, R)$.\nThe *projective line over $R$* is the set\n$$\\mathrm{P}(R) = \\set{U(a, b) : (a, b) \\in R^2 \\text{ admissible}},$$\nwhere $U(a, b) = \\set{(ua, ub) : u \\in U}$.\n\n## Remarks\n\nThe group $\\mathrm{GL}(2, R)$ acts on $\\mathrm{P}(R)$ by $U(a, b) \\mapsto U\\big((a, b) M\\big)$.\nIf $R$ is commutative, $(a, b)$ is admissible if and only if $aR + bR = R$.\nIndeed, if $ax + by = 1$ then $\\begin{pmatrix} a \u0026 b \\\\ -y \u0026 x \\end{pmatrix}$ has determinant $1$; conversely the first row of an invertible matrix generates $R$, by expanding its determinant.\n\nExample: if $R$ is a field $F$, every $(a, b) \\ne (0, 0)$ is admissible, and $\\mathrm{P}(F)$ is the usual projective line $\\set{U(x, 1) : x \\in F} \\cup \\set{U(1, 0)}$.\n\nExample: for $R = \\Z$, admissible means $\\gcd(a, b) = 1$ and $U = \\set{1, -1}$, so $\\mathrm{P}(\\Z)$ is in bijection with $\\Q \\cup \\set{\\infty}$ by $U(a, b) \\mapsto a / b$.\n\nNon-example: in $\\Z$, the pair $(2, 0)$ is not admissible, because $2\\Z + 0\\Z = 2\\Z \\ne \\Z$.\n",
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