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- Projective_line_over_a_ring abstract "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 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(a,b).P(A) = {U(a,b): aA + bA = A }, 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 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(a,1). 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 → u a v 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.".
- Projective_line_over_a_ring wikiPageID "1271901".
- Projective_line_over_a_ring wikiPageRevisionID "600970694".
- Projective_line_over_a_ring hasPhotoCollection Projective_line_over_a_ring.
- Projective_line_over_a_ring subject Category:Inversive_geometry.
- Projective_line_over_a_ring subject Category:Projective_geometry.
- Projective_line_over_a_ring subject Category:Ring_theory.
- Projective_line_over_a_ring comment "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 A × A are related when there is a u in U such that ua = c and ub = d. This relation is an equivalence relation.".
- Projective_line_over_a_ring label "Projective line over a ring".
- Projective_line_over_a_ring label "反転環幾何".
- Projective_line_over_a_ring sameAs 反転環幾何.
- Projective_line_over_a_ring sameAs m.04p0kz.
- Projective_line_over_a_ring sameAs Q7249472.
- Projective_line_over_a_ring sameAs Q7249472.
- Projective_line_over_a_ring wasDerivedFrom Projective_line_over_a_ring?oldid=600970694.
- Projective_line_over_a_ring isPrimaryTopicOf Projective_line_over_a_ring.