Ring and Module Hulls

The existence and usefulness of the injective hull of a module is well known. In this chapter several hulls for a ring or a module which are essential extensions that are “minimal”, in some sense, with respect to being contained in some designated class of rings or modules are introduced. The definition of hulls includes most of the known hulls (e.g., injective, quasi-injective, continuous, quasi-continuous, etc.), as well as, some relatively newer ones (e.g., quasi-Baer, right FI-extending, right p.q.-Baer, idempotent closure, right duo). The transfer of information between these hulls and their base rings or modules is discussed. In Sects. 8.1 and 8.2, basic results and examples are provided. In Sect. 8.3, the maximal right ring of quotients for any ring is shown to enjoy a generalized extending property for a particular set of ideals. A consequence of this result is that every ring has a hull in the idempotent closure class of rings. For a semiprime ring, its idempotent closure hull coincides with the quasi-Baer ring hull and the FI-extending ring hull. In the fourth section, our focus is on modules. An in-depth-treatment is given to the known results on the existence of continuous hulls. Then an FI-extending hull is shown to exist for every finitely generated projective module over a semiprime ring. Finally, in contrast to essential extensions of extending modules, both the extending and the FI-extending properties are shown to transfer from a module to its rational hull.