Some near-rings in which all ideals are intersections of Noetherian quotients. (Q1011030)

First, it is established that every ideal of a zero symmetric ring-free tame nearring with identity is dense in the intersection of the Noetherian quotients that contain it. Then, the ideal structure of certain function nearrings is studied. More precisely, let \(G\) be a group with countably many elements. The ideal lattice of the nearring of functions from \(G\) into itself which are compatible [see \textit{E. Aichinger} and \textit{P. M. Idziak}, J. Algebra 271, No. 1, 65-107 (2004; Zbl 1033.08001), for definition] with a certain lattice of normal subgroups of \(G\) is considered. In particular, let \(G\) be a finite group, \(n\geq 1\), and \(A_i\), \(i=1,\dots,n\), some normal subgroups of \(G\) with \(\{0\}=A_1<A_2<\cdots<A_n=G\) and \(|A_{i+1}/A_i|\geq 3\) (\(i\geq 2\)). Then the nearring of all zero-preserving functions from \(G\) to \(G\) which are compatible with all \(A_i\), \(1\leq i\leq n\), is subdirectly irreducible, and the ideal lattice is isomorphic to the lattice \(S(n)\). Here \(S(n)\) consists of the functions \(f\colon\{1,2,\dots,n\}\to\{1,2,\dots,n\}\) such that \(f(i)\leq f(j)\leq j\) for all \(i\leq j\), and for two \(f,g\in S(n)\), \(f\leq g\) if and only if \(f(i)\leq g(i)\) for all \(i=1,2,\dots,n\).