INTRODUCTION TO THE THEORY AND STRUCTURES OF MODULES

The concepts of module or quotient module have similar perspectives of motivations with the definition of a factor or a quotient ring. The additive abelian structure is induced by the additive structure on it. The projective modules are duals of the injective modules. Every free module is projective. This is another way of saying that the projective modules are generalizations of the free modules. Further any projective module is a direct summand of a free module. Thus the injective modules generally possess the property that every R - module is a submodule of an injective module. The major role of the infinite cyclic group is taken over by the additive group of R. This happens in a group with R as the operator ring. Suppose that R is considered as a right R - module selection can be made as generator the unit element of R or any divisor of the unit element. The direct sum of an arbitrary set of such groups will usually be called a free R - module.