Rings and Geometry

I: Non-Commutative Algebraic Geometry.- Principles of non-commutative algebraic geometry.- § 1 Free algebras and free fields.- § 2 Specializations and the rational topology.- § 3 Singularities of matrices over a free ring.- § 4 Existentially closed fields and the Nullstellensatz.- Applications of results on generalized polynomial identities in Desarguesian projective spaces.- § 1 Introduction.- § 2 Non-degenerate normal curves.- § 3 Degenerate conics.- § 4 Degenerate normal curves.- II: Hjelmslev Geometries.- A topological characterization of Hjelmslev's classical geometries.- § 1 Hjelmslev planes and Hjelmslev rings.- § 2 Construction of commutative H-rings.- § 3 The geometric significance of nilpotent radicals.- § 4 Topological Hjelmslev planes.- § 5 Locally compact H-planes.- § 6 Characterizations of commutative H-rings with nilpotent radicals.- § 7 Locally compact connected pappian Hjelmslev planes.- Finite Hjelmslev planes and Klingenberg epimorphisms.- § 1 K-structures and H-structures.- § 2 Nets and non-existence results for K-structures.- § 3 Nets and non-existence results for H-structures.- § 4 Desarguesian K-planes.- § 5 Auxiliary matrices.- § 6 Quadratic forms and a PH-plane with q2 ? q1.- § 7 Regular K-structures.- § 8 Generalizations of Singer's theorem and a recursive construction.- § 9 Eumorphisms of regular K-structures.- § 10 Balanced H-matrices.- § 11 Recursive constructions.- § 12 Open problems.- III: Geometries over Alternative Rings.- Generalizing the Moufang plane.- § 1 Inhomogeneous and homogeneous coordinates.- § 2 Collineations of real projective planes.- § 3 The real projective plane P( IR ) as a homogeneous space.- § 4 Abstract projective planes.- § 5 A Jordan algebra construction of projective planes.- § 6 The Hjelmslev-Moufang plane.- § 7 Algebraic transvections in P(O).- § 8 Axiomatization and coordinatization of P(O).- § 9 P(O) as homogeneous space.- § 10 Another realization of ?.- § 11 Jordan pairs - a final look at the Hjelmslev-Moufang plane.- § 12 Abstract Moufang-Veldkamp planes.- Projective ring planes and their homomorphisms.- A. Algebraic preliminaries.- § 1 Free modules and their subspaces.- § 2 Stable rank of a ring.- B. Projective ring planes.- § 3 The projective plane over a ring of stable rank 2.- § 4 Barbilian planes.- § 5 Collineations and affine collineations.- § 6 Barbilian transvection planes.- § 7 Projective ring planes.- § 8 Coordinatization of projective ring planes.- § 9 Projective planes over special types of rings.- C. Homomorphisms of projective ring planes.- § 10 Homomorphisms of Barbilian planes.- § 11 Distant-preserving homomorphisms.- § 12 Algebraic characterization of full incidence homomorphisms.- § 13 Full neighbor-preserving homomorphisms.- § 14 Admissible subrings.- IV: Metric Ring Geometries, Linear Groups over Rings and Coordinatization.- Topics in geometric algebra over rings.- § 1 Collineations between projective spaces.- § 2 Collineations between lines.- § 3 Non-injective maps which preserve generalized harmonic quadruples.- § 4 The structure of GLn(R).- Metric geometry over local-global commutative rings.- § 1 LG-rings.- § 2 Linear algebra.- § 3 GL (2).- § 4 Inner_product spaces and the orthogonal group.- § 5 Witt rings.- § 6 The symplectic and unitary groups.- Linear mappings of matrix rings preserving invariants.- § 1 Introduction.- § 2 The linear algebraic approach of McDonald, Marcus, and Moyls.- § 3 The group scheme approach of Waterhouse.- § 4 Concluding remarks.- Kinematic algebras and their geometries.- § 1 Motivation and historical review.- § 2 Problems resulting from classical kinematics; a survey of the material covered in this paper.- § 3 2-algebras.- § 4 Alternative kinematic algebras.- § 5 Geometric derivations of 2-algebras.- § 6 2-algebras whose projective derivation is an affine porous space.- § 7 The kinematic derivation of an alternative kinematic algebra. Represe