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## Über dieses Buch

The focus of this monograph is the study of rings and modules which have a rich supply of direct summands with respect to various extensions. The first four chapters of the book discuss rings and modules which generalize injectivity (e.g., extending modules), or for which certain annihilators become direct summands (e.g., Baer rings). Ring extensions such as matrix, polynomial, group ring, and essential extensions of rings from the aforementioned classes are considered in the next three chapters. A theory of ring and module hulls relative to a specific class of rings or modules is introduced and developed in the following two chapters. While applications of the results presented can be found throughout the book, the final chapter mainly consists of applications to algebra and functional analysis. These include obtaining characterizations of rings of quotients as direct products of prime rings and descriptions of certain C*-algebras via (quasi-)Baer rings.

Extensions of Rings and Modules introduces for the first time in book form:

* Baer, quasi-Baer, and Rickart modules
* The theory of generalized triangular matrix rings via sets of triangulating idempotents
* A discussion of essential overrings that are not rings of quotients of a base ring and Osofsky's study on the self-injectivity of the injective hull of a ring
* Applications of the theory of quasi-Baer rings to C*-algebras

Each section of the book is enriched with examples and exercises which make this monograph useful not only for experts but also as a text for advanced graduate courses. Historical notes appear at the end of each chapter, and a list of Open Problems and Questions is provided to stimulate further research in this area.

With over 400 references, Extensions of Rings and Modules will be of interest to researchers in algebra and analysis and to advanced graduate students in mathematics.

## Inhaltsverzeichnis

### Chapter 1. Preliminaries and Basic Results

In this beginning chapter of the book, basic notions, definitions, terminology, and notations used throughout the book are presented. Preliminary results and related material have been included for the convenience of the reader.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 2. Injectivity and Some of Its Generalizations

Conditions which generalize injective modules and which relate to this notion are presented in this chapter. The notions of extending, quasi-continuous and continuous modules are discussed and a number of their properties are included. It is known that direct summands of modules satisfying either of (C1), (C2) or (C3) conditions, of (quasi-)injective modules, or of (quasi-)continuous modules inherit these respective properties. On the other hand, these classes of modules are generally not closed under direct sums. One focus of this chapter is to discuss conditions which ensure that such classes of modules are also closed under direct sums. Applications of these notions, which include decomposition theorems, are also considered. As a natural generalization of the extending property, the notion of an FI-extending module (i.e., a module for which every fully invariant submodule is essential in a direct summand) is presented. A longstanding open problem is the precise characterization of when a direct sum of extending modules is extending. Using the FI-extending property, one can see that an arbitrary direct sum of extending modules satisfies at least the extending property for its fully invariant submodules without any additional conditions. The closely related notion of strongly FI-extending modules is also introduced. Properties of FI-extending and strongly FI-extending modules are discussed.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 3. Baer, Rickart, and Quasi-Baer Rings

This chapter is devoted to properties of rings for which certain annihilators are direct summands. Such classes of rings include those of Baer rings, right Rickart rings, quasi-Baer rings, and right p.q.-Baer rings. The results and the material presented in this chapter will be instrumental in developing the subject of our study in later chapters. It is shown that the Baer and the Rickart properties of rings do not transfer to the rings of matrices or to the polynomial ring extensions, while the quasi-Baer and the p.q.-Baer properties of rings do so. The notions of Baer and Rickart rings are compared and contrasted in Sect. 3.1 and the notions of quasi-Baer and principally quasi-Baer rings in Sect. 3.2, respectively. A result of Chatters and Khuri shows that there are strong bonds between the extending and the Baer properties of rings. We shall also see some instances where the two notions coincide. It is shown that there are close connections between the FI-extending and the quasi-Baer properties for rings.
One of the motivations for the study of the quasi-Baer and p.q.-Baer rings is the fact that they behave better with respect to various extensions than the Baer and Rickart rings. For example, as shown in this chapter, each of the quasi-Baer and the p.q.-Baer properties is Morita invariant. This useful behavior will be effectively applied in later chapters. The results on the transference (or the lack of transference) of the properties presented in this chapter to matrix and polynomial ring extensions are intended to motivate further investigations on when these properties transfer to various other extensions.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 4. Baer, Quasi-Baer Modules, and Their Applications

The Baer and the quasi-Baer properties of rings are extended to a module theoretic setting in this chapter. Using the endomorphism ring of a module, the notions of Baer, quasi-Baer, and Rickart modules are introduced and studied. Similar to the fact that every Baer ring is nonsingular, we shall see that every Baer module satisfies a weaker notion of nonsingularity of modules ($$\mathcal{K}$$-nonsingularity) which depends on the endomorphism ring of the module. Strong connections between a Baer module and an extending module will be observed via this weak nonsingularity and its dual notion. It is shown that an extending module which is $$\mathcal{K}$$-nonsingular is precisely a $$\mathcal{K}$$-cononsingular Baer module. This provides a module theoretic analogue of the Chatters-Khuri theorem for rings. Direct summands of Baer and quasi-Baer modules respectively inherit these properties. This provides a rich source of examples of Baer and quasi-Baer modules, since one can readily see that for any (quasi-)Baer ring R and an idempotent e in R, the right R-module eR R is always a (quasi-)Baer module. It will be seen that every projective module over a quasi-Baer ring is a quasi-Baer module. Connections of a (quasi-)Baer module and its endomorphism ring are discussed. Characterizations of classes of rings via the Baer property of certain classes of free modules over them are presented. An application also yields a type theory for $$\mathcal{K}$$-nonsingular extending (continuous) modules which, in particular, improve the type theory for nonsingular injective modules provided by Goodearl and Boyle.
Similar to the case of Baer modules, close links between quasi-Baer modules and FI-extending modules are established via a characterization connecting the two notions. The concepts of FI-$$\mathcal{K}$$-nonsingularity and FI-$$\mathcal{K}$$-cononsingularity are introduced and utilized to obtain this characterization. Analogous to right Rickart rings, the notion of Rickart modules is introduced as another application of the theory of Baer modules in the last section of the chapter. Connections of Rickart modules to their endomorphism rings are shown. A direct sum of Rickart modules is not Rickart in general. The closure of the class of Rickart modules with respect to direct sums is discussed among other recent results on this notion.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 5. Triangular Matrix Representations and Triangular Matrix Extensions

In this chapter, generalized triangular matrix representations are discussed by introducing the concept of a set of left triangulating idempotents. A criterion for a ring with a complete set of triangulating idempotents to be quasi-Baer is provided. A structure theorem for a quasi-Baer ring with a complete set of triangulating idempotents is shown using complete triangular matrix representations. A number of well known results follow as consequences of this useful structure theorem. The results which follow as a consequence include Levy’s decomposition theorem of semiprime right Goldie rings, Faith’s characterization of semiprime right FPF rings with no infinite set of central orthogonal idempotents, Gordon and Small’s characterization of piecewise domains, and Chatters’ decomposition theorem of hereditary noetherian rings. A result related to Michler’s splitting theorem for right hereditary right noetherian rings is also obtained as an application. The Baer, the quasi-Baer, the FI-extending, and the strongly FI-extending properties of (generalized) triangular matrix rings are discussed. A sheaf representation of quasi-Baer rings is obtained as an application of the results of this chapter.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 6. Matrix, Polynomial, and Group Ring Extensions

Transference of the ring properties discussed in previous chapters to various ring extensions is the focus of this chapter. Results developed earlier are utilized to do so. It is observed that the Baer property of a ring does not transfer to its ring extensions so readily and that this happens only under special conditions. However, it will be shown that the quasi-Baer property transfers to various matrix and polynomial ring extensions without any additional assumptions. An exploration of the transference of the two aforementioned properties as well as of the Rickart, extending, p.q.-Baer, and the FI-extending properties to various ring extensions of the given ring is carried out. The extensions of a ring considered, include its matrix (both finite and infinite), polynomial, Ore, and group ring extensions. A characterization of a semiprime quasi-Baer group algebra is presented as a consequence.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 7. Essential Overring Extensions-Beyond the Maximal Ring of Quotients

The focus of this chapter is on right essential overrings of a ring which are not right rings of quotients. Osofsky’s well-known example of a finite ring whose injective hull has no compatible ring structure is considered and generalized. All possible right essential overrings of the ring in Osofsky’s example are discussed. A ring R is constructed with a module essential extension S which is not the injective hull of R. However, S is shown to have one compatible ring structure which is a QF-ring and another compatible ring structure which is not even right FI-extending. Finally, Osofsky compatibility is discussed and a class of rings whose injective hulls have distinct compatible ring structures is studied.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 8. 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.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 9. Hulls of Ring Extensions

In this chapter, the theory of ring hulls is used to determine when various ring extensions are in the classes of interest (e.g., right (FI-) extending, (quasi-) Baer, etc.) or when certain subrings (e.g., the fixed ring) are in these classes. Section 9.1 begins with a characterization of a right extending ring whose maximal right ring of quotients is the 2×2 matrix ring over a division ring. This result eventually leads to a characterization of all right rings of quotients of a 2×2 upper triangular matrix ring over a commutative PID which are right extending, Baer, right Rickart, or right semihereditary. Skew group rings and fixed rings are considered in Sect. 9.2. The main results of this section concern semiprime rings with a group of X-outer ring automorphisms which have their skew group ring and/or fixed ring being quasi-Baer. In the final section, various matrix ring extensions (both finite and infinite) and monoid ring extensions of a ring hull are compared to the corresponding ring hull of the matrix or monoid ring extension. Moreover, for a semiprime ring R which is Morita equivalent to a ring S, then their quasi-Baer ring hulls are also Morita equivalent.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 10. Applications to Rings of Quotients and C∗-Algebras

The focus of the final chapter of the book is on applications of the ideas and results developed in earlier chapters to Functional Analysis and Ring Theory. The chapter begins with the development of necessary and sufficient conditions on a ring so that its maximal right ring of quotients can be decomposed into a direct products of indecomposable rings or into a direct products of prime rings. This relies on results on the idempotent closure class of rings discussed in Chap. 8 and a dimension on bimodules introduced in this chapter. As an application, a quasi-Baer ring hull of a semiprime ring with only finitely many minimal prime ideals is shown to be a finite direct sum of prime rings. Conditions for a ∗-ring to become a Baer ∗-ring or a quasi-Baer ∗-ring are discussed. The quasi-Baer ∗-ring property is shown to transfer from a ring to its various polynomial extensions. Self-adjoint ideals in Baer ∗-rings and quasi-Baer ∗-rings are examined. In applications to the study of C -algebras, it is shown that a unital C -algebras A is boundedly centrally closed if and only if A is a quasi-Baer ring. The local multiplier algebra of a C -algebras is shown to be always quasi-Baer. Characterizations of C -algebras whose local multiplier algebras are C -direct products of prime C -algebras are provided. The quasi-Baer property is discussed for a C -algebras A with a finite group G of -automorphisms in terms of the skew group ring A G and the fixed ring. Finally, C -algebras satisfying a polynomial identity with only finitely many minimal prime ideals are characterized.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Chapter 11. Open Problems and Questions

In this chapter, we present a list of open problems and questions to stimulate further research on the material discussed in this monograph.
Gary F. Birkenmeier, Jae Keol Park, S. Tariq Rizvi

### Backmatter

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