The projective line P("A") over a ring "A" can also be identified as the space of projective modules in the module formula_7. An element of P("A") is then a direct summand of formula_7. This more abstract approach follows the view of projective geometry as the geometry of subspaces of a vector space, sometimes associated with the lattice theory of Garrett Birkhoff or the book "Linear Algebra and Projective Geometry" by Reinhold Baer. In the case of the ring of rational integers Z, the module summand definition of P(Z) narrows attention to the "U"["m, n"], "m" coprime to "n", and sheds the embeddings which are a principle feature of P("A") when "A" is topological. The 1981 article by W. Benz, Hans-Joachim Samaga, & Helmut Scheaffer mentions the direct summand definition.