Title:

Some special classes of modules

The aim of this report is to study rings whose module classes have special properties and modules which subgenerate module classes with special properties. Of the four main chapters, three concern classes where every member has a particular type of decomposition, while the other one concerns classes which are closed under carrying out particular operations. In each case, we will try and relate the property of the class to the submodule or ideal structure of the module or ring which induces it. Chapter 2 is about a class of rings called the right coH rings, which generalise the QF rings. Like QF rings, these can be classified in numerous different ways, the original four of which were shown to be equivalent by Oshiro in [34]. The original Theorem is reproduced here as Theorem 2.1.5. In this chapter, we show a new, shorter way to prove Oshiro's Theorem using a new result, Lemma 2.2.7. We also expand Oshiro's example from [34], and construct some concrete examples of rings which are right coH but neither QF nor right H (the H property being a dual of the coH property). At the end of the chapter there is a short discussion about whether it is possible to weaken the rather strong conditions required by Lemma 2.2.7. Chapter 3 concerns the class sigma[M] of modules subgenerated by the module M. For an arbitrary M, we can only say that sigma[M] is closed under the operations of taking submodules, factor modules and direct sums. We will be asking which modules subgenerate classes which are closed under the operations of taking extensions and taking essential extensions, and trying to find out about their submodule structure. We are able to characterise these modules completely in two cases; (1) Where the annihilator of the module is the annihilator of a finite subset of the module (Corollary 3.2.16 and Corollary 3.2.17). (2) Where the modules are indecomposable injectives over a commutative noetherian ring (Theorem 3.3.16). We are also able to show a few other partial results. In Chapter 4, we wonder whether we can classify rings whose right modules are direct sums of uniform modules. In the first part of the Chapter, we show some basic results on these rings. The main part of the Chapter is taken up by the construction of examples which suggest that a complete classification may be impossible. Chapter 5 asks a generalisation of the central question of Chapter 4  which modules M have the property that every module in sigma[M] is a direct sum of uniform modules? In the case where the annihilator of M is the annihilator of a finite subset of M, we are able to provide a complete answer or at least an answer in terms of the (unclassifiable?) rings of Chapter 4. In the case where the ring is commutative, we show that our property holds if and only if M is a puresemisimple module in the sense of [52] (Theorem 5.2.4). To illustrate the topics under discussion, lots of examples have been included, particularly in Chapter 4 where they are the inspiration for the chapter's conclusions. The aim is for this report to be as selfcontained as possible and with this in mind proofs of some known results have been included. Also, for the sake of completeness, on occasion we will prove a result directly rather than deducing it from a known result not included in the report. In Chapters 4 and 5, however, we are forced to use some results which we will not prove, since some of these proofs are heavily dependent on theory which we do not have space to introduce.
