N-series and tame near-rings

Abstract

Synopsis Let N be a zero-symmetric near-ring with identity and let G be an N -group. We consider in this paper nilpotent ideals of N and N -series of G and we seek to link these two ideas by defining characterizing series for nilpotent ideals. These often exist and in most cases a minimal characterizing series exists. Another special N -series is a radical series, that is a shortest N -series with a maximal annihilator. These are linked to appropriate characterizing series. We apply these ideas to obtain characterizing series for the radical of a tame near-ring N , and to show that these exist if either G has both chain conditions on N -ideals or N has the descending chain condition on right ideals. In the latter case this provides a new proof of the nilpotency of the radical of a tame near-ring with DCCR, and an internal method for constructing minimal and maximal characterizing series for the radical.