You don’t need a ring, you need a conical group, that is a group with a one-parameter of contracting automorphisms. This type of generalization of affine geometry following Artin style has been done here: Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91–136.

You don’t need a ring, you need a conical group, that is a group with a one-parameter of contracting automorphisms. This type of generalization of affine geometry following Artin style has been done here: Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91–136.

You don´t need a ring, you need a conical group, that is a group with a one-parameter of contracting automorphisms. This type of generalization of affine geometry following Artin style has been done here: Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91-136

You don’t need a ring, you need a conical group, that is a group with a one-parameter of contracting automorphisms. This type of generalization of affine geometry following Artin style has been done here: Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91–136.