Dual projective line

Projective geometry had become the geometric language of algebra, and projective spaces had become naturally associated with vector spaces. We use this algebraic approach to make a generalization: we associate the projective line to the free R-modules of rank two. We define the projective line and dual projective line over a ring R and prove that there exists a bijective correspondence between projective space and dual projective space. We also prove that the bilinear form associated with this bijective correspondence determines a symplectic structure over the R-module R2, where R is a total quotient ring.