The Category of f-Modules

In order to gain information about the category of f-modules it is useful to understand the free f-modules as well as the injective f-modules. Because there are generally no injectives in this category our efforts will be spent on studying those relative injectives that arise by bounding the cardinality of the f-module to which a given morphism is to be extended. Sophisticated techniques will be required to characterize these f-modules. One of the characterizing properties they have, not surprisingly, is that of being an injective module; the other properties are all order theoretic. These order properties can also be used to characterize the relative injectives in other categories of ordered structures. We will first construct the injective hull of a module and the analogous maximal right quotient ring of a ring. With an eye toward our applications we investigate the maximal right quotient ring of a semiprime ring whose Boolean algebra of annihilator ideals is atomic and certain torsion-free modules over this ring. One fundamental question that arises is to determine when the injective hull of an f-module is an f-module extension and when the maximal right quotient ring of an f-ring is an f-ring extension. The answer is given in the more general context of rings and modules of quotients with respect to a hereditary torsion theory. Large classes of po-rings are identified over which all torsion-free f-modules have this property.