Nearrings: Geneses and Applications

Hardcover | September 1, 1994

byJames R. Clay

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Nearrings arise naturally in various ways, but most nearrings studied today arise as the endomorphisms of a group or cogroup object of a category. These nearrings are rings if the group object is also a cogroup object. During the first half of the twentieth century, nearfields were formalized and applications to sharply transitive groups and to foundations of geometry were utilized. 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, the design of statistical experiments, families of mutually orthogonal Latin squares and constructingplanes with circles having radius and centre even though there is no metric involved. Even though nearrings may lack the extra symmetry of a ring, there is often a very sophisticated elegance in their structure. It has recently been observed that there is an abundance of symmetry in finite cirucular planar nearrings, which disappear if the nearring is a ring.

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Nearrings arise naturally in various ways, but most nearrings studied today arise as the endomorphisms of a group or cogroup object of a category. These nearrings are rings if the group object is also a cogroup object. During the first half of the twentieth century, nearfields were formalized and applications to sharply transitive gr...

James R. Clay is at University of Arizona, Tucson.

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Format:HardcoverDimensions:480 pages, 9.25 × 6.3 × 1.26 inPublished:September 1, 1994Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0198533985

ISBN - 13:9780198533986

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Table of Contents

Chapter 1: Introduction to nearrings1.1. Getting acquainted1.2. Lots of examples1.3. Many cheerful facts about nearringsChapter 2: Planar Nearrings2.1. Planarity for nearrings2.2. Construction of circular planar nearrings2.3. Geometry of curcular planar nearrings2.4. Other geometric structures from planar nearrings2.5. Coding, cryptography, and combinatorics2.6. Sharply transitive groups and nondesarguesian planesChapter 3: The great unifier3.1. A little category theory3.2. Group and cogroup objects3.4. ExamplesChapter 4: Some first families of nearrings and some of their ideals4.1. First. what is a nearring module?4.2. Centralizer and transformation nearrings4.3. Distributively generated nearrings4.4. The ideals of abstract affine nearrings4.5. Polynomial nearrings4.6. Power series nearringsChapter 5: Some structure of groups of units5.1. Preliminaries5.2. Direct products in groups of units5.3. Semidirect products and wreath products5.4. Group extensions with factor sets5.5. A mixture of the aboveChapter 6: Avant-garde families of nearrings6.1. Sandwich and laminated nearrings6.2. Syntactic nearrings6.3. The cornucopiaAppendix: Various diagramsList of symbolsBibliographyIndex

Editorial Reviews

`a very useful addition to the literature on near-rings ... there is the pleasant and light approach of the author which makes the material easier to read, particularly in those parts where the nature of the material makes for heavy going ... This book can be thoroughly recommended.'J.D.P. Meldrum, Besprechungsbelege, February 1994