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

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 < < A_n-1 < A_n = G be a chain of normal subgroups of G with |A_i/A_i-1| ≥ 3 for all i ∈ {2,...,n}. Then the lattice of ideals of the near-ring of zero-preserving functions compatible with A_i for all i is shown to consist entirely of intersections of Noetherian quotients. The unique minimal ideal of these near-rings is explicitly determined.