Near-rings with no non-zero nilpotent two-sided R-subsets (Q1100267)

Let R be a near-ring with no non-zero nilpotent two-sided R-subsets. Then the following results are true: (1) If the annihilator of any non-zero ideal is contained in some maximal annihilator, then R is a subdirect sum of strictly prime near-rings. (2) If R satisfies a.c.c. or d.c.c. on annihilating ideals of the form Ann(Q), where Q is an ideal of R, then R is a finite subdirect sum of strictly prime near-rings. Moreover, it is also proved that if R is a regular and right duo near-ring that satisfies a.c.c. or d.c.c. on annihilating ideals of the form Ann(Q), where Q is a ideal of R, then R is a finite direct sum of near-rings \(R_ i\) (1\(\leq i\leq n)\) where each \(R_ i\) is simple and strictly prime.