---
tag: M6BN1K0B30
kind: concept
name: Projective line over a ring
msc: [51C05]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_line_over_a_ring
same_as:
  wikidata: Q7249472
  wikipedia: Projective line over a ring
---

## Summary

The projective line over a [[ring|0B1261B5ED]] $R$ extends the usual projective line over a field to any associative ring with identity.
Its 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.
The group $\mathrm{GL}(2, R)$ acts on it.
The construction gives models of circle geometries, such as the Möbius plane from the complex numbers and the Minkowski plane from the split-complex numbers.

## Explanations

### L1 human

Over a field, a point of the projective line is a ratio $a : b$ with $a$ and $b$ not both zero.
Over a ring one must be more careful, because a nonzero element need not be invertible.
The 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.
Over the integers, this gives the reduced fractions $a / b$ together with $1 / 0 = \infty$, that is, the rationals with a point at infinity.