Editing Projective line over a ring (section)

{{short description|Projective construction in ring theory}}
[[File:Projectivisation_F7P%5E1.svg|thumb|200px|Eight colors illustrate the projective line over Galois field GF(7)]]

In [[mathematics]], the '''projective line over a ring''' is an extension of the concept of [[projective line]] over a [[field (mathematics)|field]]. Given a [[ring (mathematics)|ring]] ''A'' (with 1), the projective line P<sup>1</sup>(''A'') over ''A'' consists of points identified by [[projective coordinates]].  Let ''A''<sup>×</sup> be the [[group of units]] of ''A''; pairs {{nowrap|(''a'', ''b'')}} and {{nowrap|(''c'', ''d'')}} from {{nowrap|''A'' × ''A''}} are related when there is a ''u'' in ''A''<sup>×</sup> such that {{nowrap|1=''ua'' = ''c''}} and {{nowrap|1=''ub'' = ''d''}}. This relation is an [[equivalence relation]]. A typical [[equivalence class]] is written {{nowrap|''U''[''a'', ''b'']}}.

{{nowrap|1=P<sup>1</sup>(''A'') = {{mset| ''U''[''a'', ''b''] | ''aA'' + ''bA'' {{=}} ''A'' }}}}, that is, {{nowrap|''U''[''a'', ''b'']}} is in the projective line if the [[right ideal|one-sided ideal]] generated by ''a'' and ''b'' is all of&nbsp;''A''.

The projective line P<sup>1</sup>(''A'') is equipped with a [[homography#Homography groups|group of homographies]]. The homographies are expressed through use of the [[matrix ring]] over ''A'' and its group of units ''V'' as follows: If ''c'' is in Z(''A''<sup>×</sup>), the [[center (group theory)|center]] of ''A''<sup>×</sup>, then the [[Group action (mathematics)|group action]] of matrix <math>\left(\begin{smallmatrix}c & 0 \\ 0 & c \end{smallmatrix}\right)</math> on P<sup>1</sup>(''A'') is the same as the action of the identity matrix. Such matrices represent a [[normal subgroup]] ''N'' of ''V''. The homographies of P<sup>1</sup>(''A'') correspond to elements of the [[quotient group]] {{nowrap|''V''{{hsp}}/{{hsp}}''N''}}.

P<sup>1</sup>(''A'') is considered an extension of the ring ''A'' since it contains a copy of ''A'' due to the embedding 
{{nowrap|''E'' : ''a'' → ''U''[''a'', 1]}}.  The [[multiplicative inverse]] mapping {{nowrap|''u'' → 1/''u''}}, ordinarily restricted to ''A''<sup>×</sup>, is expressed by a homography on P<sup>1</sup>(''A''):
: <math>U[a,1]\begin{pmatrix}0&1\\1&0\end{pmatrix} = U[1, a] \thicksim U[a^{-1}, 1].</math>
Furthermore, for {{nowrap|''u'',''v'' ∈ ''A''<sup>×</sup>}}, the mapping {{nowrap|''a'' → ''uav''}} can be extended to a homography:
: <math>\begin{pmatrix}u & 0 \\0 & 1 \end{pmatrix}\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} v & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} u & 0 \\ 0 & v \end{pmatrix}. </math>
: <math>U[a,1]\begin{pmatrix}v&0\\0&u\end{pmatrix} = U[av,u] \thicksim U[u^{-1}av,1].</math>
Since ''u'' is arbitrary, it may be substituted for ''u''<sup>−1</sup>.
Homographies on P<sup>1</sup>(''A'') are called '''linear-fractional transformations''' since
: <math>U[z,1] \begin{pmatrix}a&c\\b&d\end{pmatrix} = U[za+b,zc+d] \thicksim  U[(zc+d)^{-1}(za+b),1].</math>