Introduction

Although we are not prepared to define the terms ‘universal property’ and ‘categorical proof’ rigorously, nevertheless an attempt to describe these might be more informative. A module is said to satisfy a universal property if for any variable module satisfying some restrictive hypotheses, there exists a module map making a certain diagram commute. The map whose existence is given connects the variable module with the one having the universal property. Roughly speaking, theorems and their proofs which do not use elements of modules are called categorical. Such proofs proceed by manipulating modules, maps, and commutative diagrams, using the associativity of maps, and invoking either assumed hypotheses and universal properties to get the existence of new maps. Indeed, it is the abundance of such proofs which will lead us to a formal study of categories.

This chapter does not give new structure theories for rings, nor does it study special classes of rings. However, it does something just as important; it gives the basic fundamental module constructions which is standard equipment for all ring theorists. Also these module constructions and manipulations is what creates a general theorem proving ability. The emphasis is on details of proofs, not general results. Some proofs will be given elementwise as well as categorically a second time. The objective will be to develop the ability to give proofs in either of the two modes, so as to be able to exploit the strengths and weaknesses of either modes.