Prime and Minimal Ideals
The general theory on various types of primeness for ideals of a nearring is close enough to the ring-theoretical one even if, as usual, each primeness type for rings generates various non equivalent primeness types for nearrings.
The more recent types of primeness, bringing to the notions of equiprime, strongly prime, 2-primal ideals, seem to have particular interest and are introduced in detail. Each of the many primality types (see definition 5.1.1) generates, in the usual natural way, a notion of “prime” radical (as intersection of the ideals with a given type of primality), and such radicals are true radicals according to general points of view in universal algebra. So, we introduce and give a first study on this subject.
In particular, we give fundamental properties for the equiprime and the 2-primal cases. Maybe it is a surprise to see that, as it happens in commutative rings, an ideal of a 2-prime nearring is 0-prime if and only if it is c-prime.
Useful links among various types of primeness, and among primitivity and primeness are given, and cases in which certain types of primeness are equivalent are collected.
A general definition of regularity (for elements of a nearring), as introduced in [Grönewald and Olivier, 1997], is able to unify the regularity conditions studied in the third chapter, to connect various notions of regularity and primeness for giving very general results. So we introduce such notions also by giving some examples of theire use in simple applications.
It is almost mandatory to report the cases in which the famous Nöther theorem on primary decomposition of an ideal is generalized. We try to accomplish this task in a manner to also give interesting structure theorems for biregular nearrings (see definition 3.3.4 as introduced in the third chapter, where we proved simple properties on the lattice of the ideals).
Studies on the heart of a subdirectly irreducible nearring are linked to the characterization of the minimal ideals of a nearring, and we conclude by touching on touching such questions even if we are forced to set some heavy conditions for obtaining results similar to the ring-theoretical ones.
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