The projective line is the set of orbits under GL(2,"R") of the free cyclic submodule "R"(1,0) of . Extending the commutative theory of Benz, the existence of a right or left multiplicative inverse of a ring element is related to P("R") and GL(2,"R"). The Dedekind-finite property is characterized. Most significantly, representation of P("R") in a projective space over a division ring "K" is accomplished with a ("K","R")-bimodule "U" that is a left "K"-vector space and a right "R"-module. The points of P("R") are subspaces of isomorphic to their complements.