Ebook: Rings and Geometry
- Tags: Algebra
- Series: NATO ASI Series 160
- Year: 1985
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Principles of Non-Commutative Algebraic Geometry....Pages 3-37
Applications of Results on Generalized Polynomial Identities in Desarguesian Projective Spaces....Pages 39-77
Front Matter....Pages 79-79
A Topological Characterization of Hjelmslev’s Classical Geometries....Pages 81-151
Finite Hjelmslev Planes and Klingenberg Epimorphisms....Pages 153-231
Front Matter....Pages 233-233
Generalizing the Moufang Plane....Pages 235-288
Projective Ring Planes and Their Homomorphisms....Pages 289-350
Front Matter....Pages 351-351
Topics in Geometric Algebra over Rings....Pages 353-389
Metric Geometry over Local-Global Commutative Rings....Pages 391-415
Linear Mappings of Matrix Rings Preserving Invariants....Pages 417-436
Kinematic Algebras and Their Geometries....Pages 437-509
Coordinatization of Lattices....Pages 511-550
Front Matter....Pages 551-551
The Advantage of Geometric Concepts in Mathematics....Pages 553-556
Back Matter....Pages 557-567
Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Principles of Non-Commutative Algebraic Geometry....Pages 3-37
Applications of Results on Generalized Polynomial Identities in Desarguesian Projective Spaces....Pages 39-77
Front Matter....Pages 79-79
A Topological Characterization of Hjelmslev’s Classical Geometries....Pages 81-151
Finite Hjelmslev Planes and Klingenberg Epimorphisms....Pages 153-231
Front Matter....Pages 233-233
Generalizing the Moufang Plane....Pages 235-288
Projective Ring Planes and Their Homomorphisms....Pages 289-350
Front Matter....Pages 351-351
Topics in Geometric Algebra over Rings....Pages 353-389
Metric Geometry over Local-Global Commutative Rings....Pages 391-415
Linear Mappings of Matrix Rings Preserving Invariants....Pages 417-436
Kinematic Algebras and Their Geometries....Pages 437-509
Coordinatization of Lattices....Pages 511-550
Front Matter....Pages 551-551
The Advantage of Geometric Concepts in Mathematics....Pages 553-556
Back Matter....Pages 557-567
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