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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.
List of contents
I: Non-Commutative Algebraic Geometry.- Principles of non-commutative algebraic geometry.- Applications of results on generalized polynomial identities in Desarguesian projective spaces.- II: Hjelmslev Geometries.- A topological characterization of Hjelmslev's classical geometries.- Finite Hjelmslev planes and Klingenberg epimorphisms.- III: Geometries over Alternative Rings.- Generalizing the Moufang plane.- Projective ring planes and their homomorphisms.- IV: Metric Ring Geometries, Linear Groups over Rings and Coordinatization.- Topics in geometric algebra over rings.- Metric geometry over local-global commutative rings.- Linear mappings of matrix rings preserving invariants.- Kinematic algebras and their geometries.- Coordinatization of lattices.- Epilog.- The advantage of geometric concepts in mathematics.- Index of Subjects.
About the author
Prof. Dr. Karl Strambach ist seit 1972 ordentlicher Professor an der Universität Erlangen. Nach seiner Habilitation arbeitete er zunächst als Universitätsdozent und Wissenschaftlicher Rat an der Universität Tübingen sowie als Professor an der Universität Kiel. 2007 wurde ihm vonder Universität Debrecen die Ehrendoktorwürde verliehen.
