Near-rings and their modules

Abstract

After an introduction defining basic structutral aspects of near-rings, this report examines how the ring-theoretic notions of generation and cogeneration can be extended from modules over a ring to modules over a near-ring. Chapter four examines matrix near-rings and connections between the J2 and JS radicals of the near-ring and the corresponding matrix near-ring. By extending the concepts of generation and cogeneration from the ring modules to near-ring modules we are investigating how important distribution and an abelian additive structure are to these two concepts. The concept of generation faces the obstacle that the image of a near-ring module homomorphism is not necessarily a subrnodule of the image space but only a subgroup, while the sum of two subgroups need not even be a subgroup. In chapter two, generation trace and socle are defined for near-ring modules and these ideas are linked with those of the essential and module-essential subgroups. Cogeneration, dealing with kernels which are always submodules proved easier to generalise. This is discussed in chapter three together with the concept of the reject, and these ideas are Iinked to the J1/2 and J2 radicals. The duality of the ring theory case is lost. The results are less simple than in the ring theory case due to the different types of near-ring module substructures which give rise to several Jacobson-type radicals. A near-ring of matrices can be obtained from an arbitrary near-ring by regarding each rxr matrix as a mapping from Nr to Nr where N is the near-ring from which entries are taken. The argument showing that the near-ring is 2-semisimple if and only if the associated near-ring of matrices is 2-semisimple is presented and investigated in the case of s-semisimplicity. Questions arising from this report are presented in the final chapter.