Editing Projective line over a ring (section)

== Modules ==
The projective line P<sup>1</sup>(''A'') over a ring ''A'' can also be identified as the space of [[projective module]]s in the [[module (mathematics)|module]] {{nowrap|''A'' ⊕ ''A''}}. An element of P<sup>1</sup>(''A'') is then a [[direct sum of modules|direct summand]] of {{nowrap|''A'' ⊕ ''A''}}. This more abstract approach follows the view of [[projective geometry]] as the geometry of [[linear subspace|subspaces]] of a [[vector space]], sometimes associated with the [[lattice theory]] of [[Garrett Birkhoff]]<ref>Birkhoff and Maclane (1953) ''Survey of modern algebra'', pp. 293–298, or 1997 AKP Classics edition, pp. 312–317</ref> or the book ''Linear Algebra and Projective Geometry'' by [[Reinhold Baer]]. In the case of the ring of rational [[integer]]s '''Z''', the module summand definition of P<sup>1</sup>('''Z''') narrows attention to the {{nowrap|''U''[''m'', ''n'']}}, ''m'' [[coprime]] to ''n'', and sheds the embeddings that are a principal feature of P<sup>1</sup>(''A'') when ''A'' is topological. The 1981 article by W. Benz, Hans-Joachim Samaga, & Helmut Scheaffer mentions the direct summand definition.

In an article "Projective representations: projective lines over rings"<ref>{{citation |first1=A. |last1=Blunck |first2=H. |last2=Havlicek |date=2000 |title=Projective representations: projective lines over rings |journal=[[Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg]] |volume=70 |pages=287–299 |doi=10.1007/BF02940921 |mr=1809553 |arxiv=1304.0098 }}. This article uses an alternative definition of projective line over a ring that restricts elements of the projective line over '''Z''' to those of the form {{nowrap|''U''[''m'', ''n'')}} where ''m'' and ''n'' are coprime.</ref> the [[group of units]] of a [[matrix ring]] M<sub>2</sub>(''R'') and the concepts of module and [[bimodule]] are used to define a projective line over a ring. The group of units is denoted by {{nowrap|GL(2, ''R'')}}, adopting notation from the [[general linear group]], where ''R'' is usually taken to be a field.

The projective line is the set of orbits under {{nowrap|GL(2, ''R'')}} of the free cyclic [[module (mathematics)#Submodules and homomorphisms|submodule]] {{nowrap|''R''(1, 0)}} of {{nowrap|''R'' × ''R''}}. Extending the commutative theory of Benz, the existence of a right or left [[multiplicative inverse]] of a ring element is related to P<sup>1</sup>(''R'') and {{nowrap|GL(2, ''R'')}}. The [[Dedekind-infinite set#Generalizations|Dedekind-finite]] property is characterized. Most significantly, [[representation theory|representation]] of P<sup>1</sup>(''R'') in a projective space over a division ring ''K'' is accomplished with a {{nowrap|(''K'', ''R'')}}-bimodule ''U'' that is a left ''K''-vector space and a right ''R''-module. The points of P<sup>1</sup>(''R'') are subspaces of {{nowrap|P<sup>1</sup>(''K'', ''U'' × ''U'')}} isomorphic to their complements.