TY - JOUR

T1 - Some near-rings in which all ideals are intersections of Noetherian quotients

AU - Aichinger, Erhard

AU - Cannon, G. Alan

AU - Fuß, Jürgen

AU - Kabza, Lucyna

AU - Neuerburg, Kent

PY - 2008/3

Y1 - 2008/3

N2 - 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.

AB - 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.

KW - Congruence preserving functions

KW - Ideals

KW - Near-rings

KW - Noetherian quotients

UR - http://www.scopus.com/inward/record.url?scp=47849130270&partnerID=8YFLogxK

U2 - 10.1216/RMJ-2008-38-3-713

DO - 10.1216/RMJ-2008-38-3-713

M3 - Article

SN - 0035-7596

VL - 38

SP - 713

EP - 726

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

IS - 3

ER -