Some near-rings in which all ideals are intersections of Noetherian quotients

Erhard Aichinger, G. Alan Cannon, Jürgen Fuß, Lucyna Kabza, Kent Neuerburg

Research output: Contribution to journalArticlepeer-review


For every near-ring, Noetherian quotients are one source of ideals, but usually not all ideals can be obtained from such quotients. In this paper, we show that every ideal of a zero symmetric ring-free tame near-ring with identity is dense in the intersection of the Noetherian quotients that contain it. In many cases, we are able to determine the ideal lattice of the near-ring of those functions on a group that are compatible with a given subset of the set of all normal subgroups. In particular, let G be a finite group, and let {0} = A 1 < A2 < < An-1 < An = G be a chain of normal subgroups of G with |Ai/Ai-1| ≥ 3 for all i ∈ {2,...,n}. Then the lattice of ideals of the near-ring of zero-preserving functions compatible with Ai for all i is shown to consist entirely of intersections of Noetherian quotients. The unique minimal ideal of these near-rings is explicitly determined.

Original languageEnglish
Pages (from-to)713-726
Number of pages14
JournalRocky Mountain Journal of Mathematics
Issue number3
Publication statusPublished - Mar 2008


  • Congruence preserving functions
  • Ideals
  • Near-rings
  • Noetherian quotients


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