Projective line over a ring

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 projective coordinates. Let U be the group of units of A; pairs (a, b) and (c, d) from A × A are related when there is a u in U such that ua = c and ub = d. This relation is an equivalence relation. A typical equivalence class is written U. 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 projective coordinates. Let U be the group of units of A; pairs (a, b) and (c, d) from A × A are related when there is a u in U such that ua = c and ub = d. This relation is an equivalence relation. A typical equivalence class is written U. P(A) = { U : aA + bA = A }, that is, U 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 ( c 0 0 c ) {displaystyle {egin{pmatrix}c&0\0&cend{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 E : a → U. The multiplicative inverse mapping u → 1/u, ordinarily restricted to the group of units U of A, is expressed by a homography on P(A): Furthermore, for u,v ∈ U, the mapping a → uav can be extended to a homography: Since u is arbitrary, it may be substituted for u−1.Homographies on P(A) are called linear-fractional transformations since Finite rings have finite projective lines. The projective line over GF(2) has three elements: U, U, and U. Its homography group is the permutation group on these three.:29 The ring Z/3Z, or GF(3), has the elements 1, 0, and −1; its projective line has the four elements U, U, U, U since both 1 and −1 are units. The homography group on this projective line has 12 elements, also described with matrices or as permutations.:31 For a finite field GF(q), the projective line is the Galois geometry PG(1, q). J. W. P. Hirschfeld has described the harmonic tetrads in the projective lines for q = 4, 5, 7, 8, 9. Consider P(ℤ/nℤ) when n is a composite number. If p and q are distinct primes dividing n, then < p > and < q > are maximal ideals in ℤ/nZ and by Bézout's identity there are a and b in Z such that a p + b q = 1 , {displaystyle ap+bq=1,} so that U is in P(ℤ/nℤ) but it is not an image of an element under the canonical embedding. The whole of P(ℤ/nZ) is filled out by elements U [ u p , v q ] , u ≠ v , u , v ∈ U = {displaystyle U,quad u eq v,quad u,vin U=} the units of ℤ/nℤ. For examples, when n =6 there is U; when n =10 there is U. In ℤ/12ℤ, U = {1, 5, 7, 11} providing more extra points than in ℤ/6ℤ. The extra points can be associated with ℚ ⊂ ℝ ⊂ ℂ, the rationals in the extended complex upper-half plane. The group of homographies on P(ℤ/nℤ) is called a modular subgroup.