A characterization of prime radical in near-rings under chain conditions on annihilators

It has been proved that ifR is a near-ring that satisfiesd.c.c anda.c.c. on right annihilators of its righR-subsets, then the prime radicalP(r) is a nilpotent ideal. A few results are included in Author's Doctoral Dissertation at Sukhadia University, Udaipur (1983).     

The general radical theory of near-rings — answers to some open problems

It is shown that in the variety of all, not necessarily 0-symmetric near-rings, there are no non-trivial classes of near-rings which satisfy condition (F), no non-trivial (Kurosh-Amitsur) radical classes with the ADS-property and consequently no non-trivial ideal-hereditary radical classes. It is also shown that any hereditary semisimple class contains only 0-symmetric near-rings.Presented by E. Fried.AMS Subject Classification: 16Y30; 16N80.     

Medial near-rings

In this paper we discuss (left) near-rings satisfying the identities:abcd=acbd,abc=bac, orabc=acb, called medial, left permutable, right permutable near-rings, respectively. The structure of these near-rings is investigated in terms of the additive and Lie commutators and the set of nilpotent elementsN (R). For right permutable and d.g. medial near-rings we obtain a Binomial Theorem, show thatN (R) is an ideal, and characterize the simple and subdirectly irreducible near-rings. Natural examples from analysis and geometry are produced via a general construction method.