Monographs on Near-rings and Near-fields


Gunter Pilz

North-Holland Mathematical Studies 23, 1983.
MR 85h-16046
This book is a reprint of the first book ever published on nearrings. It remains in its place as the first and somehow still the standard nearring text. It contains a thorough introduction to the theory of nearrings, along with some applications and diversions into nearfields and other related subjects.

A rough summary would be as follows: Nearrings are a natural object of study as algebraic objects, relating to many other abstract algebraic entities. Many interesting structural results are brought out, and the geneology ofthe results is made quite clear. Some connections to geometry with planar nearrings are brought out, as are the important aspects of nearfield theory, especially as it relates to the concept of a neardomain. Lastly some applications of nearrings are considered.

A comprehensive table of remarkable examples and counter examples, a collection of simple and not so simple research problems, and a comprehensive collection of examples round out this fine monograph.

Near-rings and their links with groups

John Meldrum

Pitman Research Notes in Mathematics 134, 1985.
MR 88a-16068
This book with it's focus on more direct group theory reaches somewhat beyond the boundaries of pure nearring theory. As the author says in his preface, it is accessible even to the final year undergraduate or beginning postgraduate, and is not as comprehensive as the Pilz book. Not to say that it is merely an introduction though! The first part of the book deals with general nearring theory, looking at the module theory, primitivity, radicals and other areas of interest in the general structure theory. The second part looks more at the group-theoretic connections of nearring theory, in particular the endomorphism nearrings and the distributively generated nearrings.

Theorie der Fastkörper

Heinz Wähling

Thales Verlag, 1987.
MR 90e-12024
Though technically not a nearring book, this belongs here due to the long-standing history of relevance between nearring and nearfield theory. The text is in German, with a nice big English summary at the end, an index and a bibliography.

Near-rings: Theory and Genesis

Jim Clay

Oxford University Press, 1992.
MR 94b-16001
This book has a very combinatorist point of view. In particular the author's work on block designs from planar nearrings plays an important role.

Jim also claims he saved 11 pages of text by omitting the hyphen to move from near-ring to nearring!

Nearrings. Some Developments Linked to Semigroups and Groups

Celestina Cotti Ferrero and Giovanni Ferrero

Kluwer Academic Publishers, Dordrecht, ISBN 1-4020-0875-9, October 2002.
A collection of many topics that near-ringers have been interested in, with many suggestions arousing curisosity to read and investigate further.
Maintained by Erhard Aichinger. Last updated 17 Dec 2002.