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  "body": "\n## Summary\n\nThe projective line over a [[ring|0B1261B5ED]] $R$ extends the usual projective line over a field to any associative ring with identity.\nIts points are the pairs $(a, b) \\in R^2$ that can be completed to an invertible $2 \\times 2$ matrix, taken up to multiplication on the left by units.\nThe group $\\mathrm{GL}(2, R)$ acts on it.\nThe construction gives models of circle geometries, such as the Möbius plane from the complex numbers and the Minkowski plane from the split-complex numbers.\n\n## Explanations\n\n### L1 human\n\nOver a field, a point of the projective line is a ratio $a : b$ with $a$ and $b$ not both zero.\nOver a ring one must be more careful, because a nonzero element need not be invertible.\nThe rule is that $(a, b)$ must be the first row of an invertible matrix, and two pairs give the same point when one is a unit times the other.\nOver the integers, this gives the reduced fractions $a / b$ together with $1 / 0 = \\infty$, that is, the rationals with a point at infinity.\n",
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