Left modules for left nearrings.

• Main Author: Grainger, Gary Ross.
• Language: en
• Published: The University of Arizona. 1988
• Subjects: Near-rings.
• Online Access: http://hdl.handle.net/10150/184480

Every ring has both left and right modules. In the theory of nearrings, only right modules are usually considered for left nearrings. The purpose of this report is to promote the study of an alternative type of nearring module. For left nearrings, these unusual modules are left modules. There are three reasons for studying left modules for left nearrings. These unusual nearring modules add an element of symmetry to the theory of nearrings. At the same time, comparing left and right modules of a left nearring illustrates how the theory of nearrings is distinct from ring theory. Finally, with two types of nearring modules, it is possible to carry over to nearring theory more concepts from ring theory; for example, duals of modules and bimodules. This report is an attempt to show that these reasons are valid. The first chapter is devoted to producing a well-reasoned definition for the unusual type of nearring module. It begins with a careful presentation of background material on nearrings, rings, and ring modules. This material is used to motivate the definitions for nearring modules, which are introduced in the third section. The second chapter is concerned with showing that the unusual type of nearring module can fit into the theory of nearrings. In the first section, several papers relevant to the study of these modules are summarized. The work of A. Frohlich on free additions is of primary importance. General construction methods for both types of nearring modules are then described. Finally, some general properties of left modules of left nearrings are examined. Examples of left modules for left nearrings are presented in the third chapter. First, the general constructions of the second chapter are applied in some particular cases. This leads naturally to structures that are analogous to bimodules and structures analogous to dual modules for ring modules. Here, free additions have a special role. Several dual nearring modules are examined in detail. The information needed to construct many more examples of nearring modules of the unusual type is also presented. Only small cyclic groups are used for these examples.