Near-Rings and Near-Fields pp 13-29 | Cite as

# Localized Distributivity Conditions

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## 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* _{1}t + *sk* _{2}t, for each *k* _{ 1 }, *k* _{ 2 } ∈ *K*, *s* ∈ *S*, *t* ∈ *T*. 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## Preview

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