## Abstract

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} < A_{2} < < 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.

Original language | English |
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Pages (from-to) | 713-726 |

Number of pages | 14 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 38 |

Issue number | 3 |

DOIs | |

Publication status | Published - Mar 2008 |

## Keywords

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