Nearrings Geneses and Applications [Hardcover]

Although Nearrings arise naturally in various ways, most nearrings studied today arise as the endomorphisms of a group or cogroup object of a category. During the first half of the twentieth century, nearfields were formalized using applications to sharply transitive groups and to foundations of geometry. This book details the theoretical implications of how planar nearrings grew out of the geometric success of the planar nearfields and have found numerous applications to various branches of mathematics as well as to coding theory, cryptography, and the design of statistical families of mutually orthogonal Latin squares and constructive planes. As the author here illustrates, nearrings may lack the extra symmetry of a ring but there is often a very sophisticated elegance in their structure and, in finite circular planar nearrings, an abundance of symmetry.

Chapter 1: Introduction to Nearrings
1.1. Getting Acquainted
1.2. Lots of Examples
1.3. Many Cheerful Facts about Nearrings
Chapter 2: Planar Nearrings
2.1. Planarity for Nearrings
2.2. Construction of Circular Planar Nearrings
2.3. Geometry of Circular Planar Nearrings
2.4. Other Geometric Structures from Planar Nearrings
2.5. Coding, Cryptography, and Combinatorics
2.6. Sharply Transitive Groups and Nondesarguesian
Chapter 3: The Great Unifier
3.1. A Little Category Theory
3.2. Group and Cogroup Objects
3.4. Examples
Chapter 4: Some First Families of Nearrings and Some of Their Ideals
4.1. First. What is a Nearring Module?
4.2. Centralizer and Transformation Nearrings
4.3. Distributively Generated Nearrings
4.4. The Ideals of Abstract Affine Nearrings
4.5. Polynomial Nearrings
4.6. Power Series Nearrings
Chapter 5: Some Structure of Groups of Units
5.1. Preliminaries
5.2. Direct Products in Groups of Units
5.3. Semidirect Products and Wreath Prol£)