Completely Prime Ideals and Radicals in Near-Rings
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An ideal I of a near-ring R is 2-primal if the prime radical of R/I equals the set of nilpotent elements of R/I We show that if \(IR\subseteq I\), then I is a 2-primal ideal of R if and only if each minimal prime ideal containing I is a completely prime ideal. A complete classification of the subdirectly irreducible zero symmetric near-rings with a 2-primal heart is provided. Zero symmetric near-rings with each prime ideal completely prime are classified in terms of the 2-primal condition. Various chain conditions are invoked on 2-primal near-rings to obtain decompositions and additive group information.
KeywordsPrime Ideal Prime Radical Homomorphic Image Nilpotent Element Matrix Ring
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