On maximal ideals of tame near-rings

Let $N$ be a zero-symmetric near-ring with identity, and let $\Gamma$ be a faithful tame $N$-group. We prove that every maximal ideal of $N$ is either dense in $N$ or equal to the annihilator of a section in the submodule lattice of $\Gamma$. We study the case that there is precisely one maximal ideal: often this maximal ideal has to be 0. As a consequence, we see that if the near-ring of zero-preserving polynomial functions on a finite $\Omega$-group $V$ has precisely one maximal ideal, then $V$ is either simple or nilpotent. Finally, we look at groups $G$ for which the near-rings $I(G)$, $A(G)$, and $E(G)$ have precisely one maximal ideal, or are even simple.