Rings and Geometry

When looking for applications of ring theory in geometry, one first thinks of algebraic geometry, which sometimes may even be interpreted as the concrete side of commutative algebra. However, this highly de­ veloped branch of mathematics has been dealt with in a variety of mono­ graphs, so that - in spite of its technical complexity - it can be regarded as relatively well accessible. While in the last 120 years algebraic geometry has again and again attracted concentrated interes- which right now has reached a peak once more - , the numerous other applications of ring theory in geometry have not been assembled in a textbook and are scattered in many papers throughout the literature, which makes it hard for them to emerge from the shadow of the brilliant theory of algebraic geometry. It is the aim of these proceedings to give a unifying presentation of those geometrical applications of ring theo~y outside of algebraic geometry, and to show that they offer a considerable wealth of beauti­ ful ideas, too. Furthermore it becomes apparent that there are natural connections to many branches of modern mathematics, e. g. to the theory of (algebraic) groups and of Jordan algebras, and to combinatorics. To make these remarks more precise, we will now give a description of the contents. In the first chapter, an approach towards a theory of non-commutative algebraic geometry is attempted from two different points of view.

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. Representation theorem.- § 8 Kinematic algebras with an adjoint map. The general notation of a kinematic map.- § 9 Kustaanheimo’s kinematic model of the hyperbolic space.- Coordinatization of lattices.- § 1 Introduction.- § 2 Basic definitions and notations.- § 3 The axioms and formulation of the coordinatization theorem.- § 4 Lemmata.- § 5 Proof of the coordinatization theorem.- § 6 A different approach.- § 7 The independence of the axioms.- Epilog.- The advantage of geometric concepts in mathematics.- Index of Subjects.