The Theory of Bases for Modules

In this paper, the principal problem is a comparison of vector spaces with modules over different kinds of rings, especially with respect to their generating systems. In the first part, a general introduction to the theory of modules and vector spaces will be presented, followed by statements of several theorems on vector spaces. Examples will then be given in the second part to show that most of the theorems fail to hold for modules in general. In the third part, possible definitions for bases will be examined, followed by a study of the results of the various definitions on the theory of modules, compared with the theory of vector spaces. In this part it will be seen to be necessary to consider certain restrictions on the module, in order to achieve significant results