Regularities and More

Abstract

This chapter is devoted to nearrings N fulfilling regularity conditions (as a ∈ aNa),variants and generalizations: they are obviously semigroup­theoretical conditions arising by classical studies on rings. As usual, we will try to see if (and how) it is possible to bring ring-theoretical results to nearring theory. It is easy to think that there are many variants of this type of idea, and each variant can be very interesting, or not so useful, depending on the context of our study.

We introduce some properties of idempotent elements of a nearring N (with or without an identity u), depending on them belong or do not belong to the multiplicative center of N. We also introduce Pierce decompositions and some lifting of idempotents (recalling classical lift­ing for groups and rings) that are maybe able to suggest semidirect decompositions of particular extensions of the Schreier type, but we do not develop it to remain simple.

Properties of reduced nearrings (if necessary fulfilling other conditions), usually dispersed throughout many papers according to the needs of each paper are collected. In this case other results on idempotents are given.

Several variants of typical regularity conditions are introduced (main­ly in definition 3.3.4) trying to fix a terminology (sometimes creating terms that may be useful to quickly recall important links between such conditions), and warnings on the different terms used by various authors treating this subject.

For example, links between right strongly regular, left strongly regular and regular nearrings are stressed. Obviously, to deepen the study on the structure of regular and strongly regular nearrings we use several ideas from the previous chapters.

Now it is possible to introduce structure theorems for regular or strongly regular nearrings both for the zero-symmetric case and for general cases. For nearrings fulfilling some suitable chain conditions we have sharper structural results.

Particular classes of regular nearrings are introduced as, for example, ortodox nearrings (i.e. nearrings in which the set of idempotents is a semigroup) and as generalized nearfields (i.e. nearrings in which the multiplicative semigroup is inverse: ∀a ∈ N there exists a unique b ∈ N such that a = aba and b = bab). A class of nearrings containing the class of generalized nearfields is the class of biregular nearrings (a nearring N is biregular if for all a ∈ N there exists a central idempotent e such that <a> _N = eN). Proving that this class is the class of Betsch’s biregular nearrings we have a structure theorem for such nearrings.

Some variants of nearrings fulfilling regularity conditions for stable (left or right) and right (left) bipotent nearrings (i.e. nearrings N fulfilling conditions as Na = aNa or aN = a ² N for all a ∈ N) are introduced. An interesting construction of stable nearrings is given using an idea of “semidirect sum” introduced in chapter II.

A tiny study of right bipotent nearrings fulfilling some chain conditions is given.

To end the chapter, we collect various cases in which regularity con­ditions or other semigroup-theoretical conditions on a nearring N force N to be a nearfield.

A girl has to kiss a lot of frogs before one of them turns into a prince.

—Anonymous (1600–2001)