Localized Distributivity Conditions

Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 336)

Abstract

Various conditions have been used in lieu of full two-sided distributivity to obtain structure theory for near-rings, most successful and most widely known being the distributively generated condition, but also including a plethora of other conditions, some being quite exotic. Imperative in justifying the use of such a condition is not only that it gives satisfory results, but that also there is a wide class of natural examples which satisfy the condition. Definition: Let K, S, and T be subsets of a (left) near-ring R, with K and T non-empty. We say “K is (S, T)-distributive” if s (k 1 +k 2 )t = sk 1t + sk 2t, for each k 1 , k 2 K, sS, tT. We say “K is (S,T)-d.g.on X” if K is (S, X)-distributive and T is contained in the subgroup of R which is generated additively by X. Generically we call such conditions “localized distributivity conditions”. First we give a historically oriented survey of such conditions and how they were used. This serves as one source of motivating examples. Then certain localized distributivity conditions on special R-subgroups or ideals are discussed with applications to the study of minimal ideals, various types of prime ideals and their associated radicals, and primitivity and the J v radicals. Examples of near-rings of mappings or homomorphisms on additive groups are given to illustrate how these particular conditions discussed arise naturally, thus satisfying both justification criteria.

Keywords

Prime Ideal Prime Ring Prime Radical Minimal Ideal Finite Simple Group
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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