Prime ideals and the ideal-radical of a distributively generated near-ring

The concepts of a prime ideal of a distributively generated (d.g.) nearring R, a prime d.g. near-ring and an irreducible R-group are introduced1). The annihilating ideal of an irreducible R-group with an R-generator is a prime ideal. Consequently we define a prime ideal to be primitively prime if it is the annihilating ideal of such an R-group, and a d.g. near-ring to be a primitively prime near-ring if it acts faithfully on such a group. The intersection of all the primitively prime ideals of a d.g. near-ring is called the ideal-radical; this ideal contains all the nilpotent ideals of the near-ring and a relationship between it and the quasi-radical of the near-ring is established. In section 2 we consider d.g. near-rings R which satisfy the descending chain condition for left R-modules. In this case, the ideal-radical is nilpotent. Any non-zero prime d.g. near-ring is a primitively prime d.g. near-ring. All irreducible R-groups with an R-generator of a non-zero prime d.g. near-ring R are shown to be isomorphic to the finite number of direct summands of the group R + N, where N is the quasi-radical of R. If R has finite order, then it has, to within an isomorphism, but one faithful representation on an irreducible R-group with an R-generator and all its irreducible R-groups with R-generators are homomorphic images of R-subgroups of this group. In section 3, a number of equivalent conditions is given for a d.g. near-ring to have a nilpotent radical. One of them is that all its proper prime ideals are maximal ideals. In section 4, we construct an example of a finite d.g. near-ring whose radical is not nilpotent and whose quasi-radical is not an ideal.