Classes of “Simpler” Nearrings
In this chapter, we collect some problems such that the study of them brings us, in a very natural way, to use theorem 2.2.23 and its variations.
Maybe (in contrast with the well-known situation for rings), there are not any nontrivial groups hosting (as additive groups) only trivial nearrings. Today the question is still not close completely, but simple applications of some constructions of the second chapter give very wide classes of groups hosting nontrivial nearrings.
Finite nearrings without proper subrings are studied in detail under the name of “strictly simple” nearrings. Note that, among the nearrings without proper ideals, we find as “trivial” cases the zero-symmetric nearrings that are just the usual (additive) simple groups equipped with a trivial product, so we do not touch on this study.
A generalization of the idea of strictly simple nearring are the p-singular nearrings, i.e. the nearrings generated by each of its elements of characteristic p (p prime). Such nearrings are studied and classified in detail also using the radical J2, but a lot of interesting problems on the subject are (often only implicitly) suggested in the main part of this chapter, and we hope to see someday other partial solutions of them.
The other (more or even too wide) generalizations of strictly simple nearrings studied in this chapter are the nearrings with a small number of subnearrings or of ideals and the nearrings generated by a small number of its elements. Once again, contrasting with classical structures, a nearring generated by one element can be very complex.
In the same framework we introduce weakly divisible nearrings because the lattice of their N-subgroups is a chain.
A study on a generalization of integrality (the case in which N contain an integral subset H such that N2 ⫅ H: then N2 is integral) is introduced, and constructions of some of these nearrings are given.
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