ERHARD AICHINGER

On maximal ideals of tame near-rings

Pages 215-233
Received: 10 October 1999
Mathematics Subject Classification (2000): 16Y30

Abstract Let N be a zero-symmetric near-ring with identity, and let G 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 G. 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 W-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.