Abstract
A new axiomatization involving incidence and remoteness of planes with nondivision coordinate rings is introduced and a coordinatization theorem is obtained. A geometric process of splitting points and lines to obtain another plane with the same coordinates is described. It is also shown that a group of Steinberg type is parametrized by a nonassociative ring. The notion of elementary basis sets for an associative ring is introduced and constructions of projective and affine planes are given. A plane with reflections determining a system of rotations is shown to have commutative, associative coordinates.
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