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Near-Ring Radicals and Class Pairs

L. Godloza

Department of Mathematics, University of Transkei Umtata 5117, South Africa

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N. J. Groenewald

Department of Mathematics and Applied Mathematics, University of Port Eklizabeth, Port Elizabeth 6000, South Africa

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W. A. Olivier

Department of Mathematics and Applied Mathematics, University of Port Eklizabeth, Port Elizabeth 6000, South Africa

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https://doi.org/10.1142/S100538670500009XCited by:0 (Source: Crossref)

Abstract

For near-ring ideal mappings ρ₁ and ρ₂, we investigate radical theoretical properties of and the relationship among the class pairs (ρ₁: ρ₂), and (ℛ_ρ₂: ℛ_ρ₁). Conditions on ρ₁ and ρ₂ are given for a general class pair to form a radical class of various types. These types include the Plotkin and KA-radical varieties. A number of examples are shown to motivate the suitability of the theory of Hoehnke-radicals over KA-radicals when radical pairs of near-rings are studied. In particular, it is shown that forms a KA-radical class, where denotes the class of completely prime near-rings and the class of 3-prime near-rings. This gives another near-ring generalization of the 2-primal ring concept. The theory of radical pairs are also used to show that in general the class of 3-semiprime near-rings is not the semisimple class of the 3-prime radical.

Keywords:

AMSC: 16Y30, 16N80

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