in mathematics, the projective line over a ring is an extension of the concept of projective line over a field. given a ring a with 1, the projective line p(a) over a consists of points identified by homogeneous coordinates. let u be the group of units of a; pairs (a,b) and (c,d) from  are related when there is a u in u such that  and . this relation is an equivalence relation. a typical equivalence class is written u(a,b).  that is, u(a,b) is in the projective line if the ideal generated by a and b is all of a. the projective line p(a) is equipped with a 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(u), the center of u, then the group action of matrix \begin{pmatrix}c & 0 \\ 0 & c \end{pmatrix} on p(a) is the same as the action of the identity matrix. such matrices represent a normal subgroup n of v. the homographies of p(a) correspond to elements of the quotient group v / n . p(a) is considered an extension of the ring a since it contains a copy of a due to the embedding . the multiplicative inverse mapping , ordinarily restricted to the group of units u of a, is expressed by a homography on p(a): :u(a,1)\begin{pmatrix}0&1\\1&0\end{pmatrix} = u(1,a) \thicksim u(a^{-1},1). furthermore, for u,v â u the mapping  can be extended to a homography: :\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}. :u(a,1)\begin{pmatrix}v&0\\0&u\end{pmatrix} = u(av,u) \thicksim u(u^{-1}av,1). since u is arbitrary, it may be substituted for uâ1. homographies on p(a) are called linear-fractional transformations since :u(z,1) \begin{pmatrix}a&c\\b&d\end{pmatrix} = u(za+b,zc+d) \thicksim u((zc+d)^{-1}(za+b),1).. ==instances== finite rings have finite projective lines. the projective line over gf(2) has three elements: u(0,1), u(1,0), and u(1,1). its homography group is the permutation group on these three.robert alexander rankin (1977) modular forms and functions, page 29, cambridge university press isbn 0-521-21212-x the ring z/3z, or gf(3), has the elements 1, 0, and â1; its projective
Algebraic_geometry Projective_geometry Ring_theory