##
**Module theory, extending modules and generalizations. With the cooperation of Patrick F. Smith.**
*(English)*
Zbl 1368.16002

Frontiers in Mathematics. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0950-4/pbk; 978-3-0348-0952-8/ebook). xx, 369 p. (2016).

Let \(R\) be an associative ring with identity. A left \(R\)-module \(M\) is called extending (or a CS-module, \(C_1\)-module) if every submodule of \(M\) is essential in a direct summand or, equivalently, every complement submodule is a direct summand of \(M\). The extending condition for modules and its dual “lifting” were firstly used by M. Harada [Osaka J. Math. 19, 189–201 (1982; Zbl 0491.16025); ibid. 19, 203–215 (1982; Zbl 0491.16026)] and K. Oshiro [Hokkaido Math. J. 13, 339–346 (1984; Zbl 0559.16012)]. The importance of extending modules and rings in ring theory and module theory became obvious in the 1990’s, but not exclusively, through the impact of the publication of monographs of S. H. Mohamed and B. J. Müller [Continuous and discrete modules. Cambridge etc.: Cambridge University Press (1990; Zbl 0701.16001)], and of Nguyen Viet Dung et al. [Extending modules. Harlow: Longman Scientific & Technical (1994; Zbl 0841.16001)]. Since that time, there has been a continuing interest in such rings and modules and their various generalizations which arose not only directly from the study of CS modules, but also from work concerning the dual notion to extending, namely, lifting modules. Many results on extending modules and rings were transferred to lifting modules and rings. At the first glance, the extending and generalized extending concepts appear to be too similar to expect many differences in their application to the structure theory of rings and modules. However, as the authors of this book indicate, there are many “surprising” differences. To this end, they classify generalized extending modules into two groups. The first group consists of generalized extending modules (called inner type generalization) such that either for every submodule or a kind of special submodules, there exists a direct summand with the property that the direct sum of the after mentioned submodules with the direct summand is essentially contained in the module. The second group consists of generalizations of extending modules (called outer type generalization) which are based on a technical condition like a homomorphism into a direct summand or an equivalence relation in the lattice of submodules, etc. They also apply their former equivalence relation idea to the dual extending case, and believe that the application will foster research on dual extending modules and related classes of modules.

The purpose of this monograph is “to give an up-to-date presentation of known and new results on generalized extending modules and some complementary results on extending matrix rings, and also provide standard background material, but with somewhat selective topics, on ring and module theory”. A number of open research problems are listed at the end of the book to generate interest in research on generalized extending and related duals. Each section includes exercises of varying degrees of difficulties for graduate students.

The book is divided into seven chapters with an appendix. The first one is an introduction to module and ring theory. The second chapter centers on certain type of modules, including those possessing the \(C_2\) and \(C_3\) properties, the summand intersection property, and the summand sum property conditions. The third chapter collects several results on extending modules and rings and continuous, quasi-continuous modules, most of which cannot be found in other monographs. The next two chapters are the essential part of the book. The fourth chapter is devoted to the study of so-called inner generalization of extending modules, the class of generalized extending modules such that either for every submodule or a kind of special submodules, there exists a direct summand with the property that the direct sum of the after mentioned submodules with the direct summand is essentially contained in the module. The authors introduce various generalization of extending modules of this type, including weak CS-modules; \(C_{11}\)-modules, \(C_{11}\)-rings, and introduce a framework which encompasses most of the generalizations of the CS property and allow them to target specific sets of submodules of a module for application of the CS property. The fifth chapter is devoting to the study on the so-called outer generalization of extending modules, the class of generalized extending modules, which are based on a technical condition like a homomorphism into a direct summand or an equivalence relation in the lattice of submodules, etc. The authors introduce and study various generalization of extending modules, including \(C_{12}\)-modules, GLS-modules, G-extending modules, and so on. Chapter 6 studies the dual Goldie and ec-complement versions of the extending property. Chapter 7 formulates open problems and questions. The problems are legitimate and provide a basis for further research which in turn will greatly broaden the scope of the theory. The appendix of the book contains a construction of rings of quotients via Gabriel topologies. The book ends with a quite complete bibliography.

The book is addressed to students and researchers in module and ring theory. It will undoubtedly be a valuable tool for everybody working in this area. Finally, it should be mentioned that the authors have made a well-thought-out effort to improve and refine the terminology on the subject.

The purpose of this monograph is “to give an up-to-date presentation of known and new results on generalized extending modules and some complementary results on extending matrix rings, and also provide standard background material, but with somewhat selective topics, on ring and module theory”. A number of open research problems are listed at the end of the book to generate interest in research on generalized extending and related duals. Each section includes exercises of varying degrees of difficulties for graduate students.

The book is divided into seven chapters with an appendix. The first one is an introduction to module and ring theory. The second chapter centers on certain type of modules, including those possessing the \(C_2\) and \(C_3\) properties, the summand intersection property, and the summand sum property conditions. The third chapter collects several results on extending modules and rings and continuous, quasi-continuous modules, most of which cannot be found in other monographs. The next two chapters are the essential part of the book. The fourth chapter is devoted to the study of so-called inner generalization of extending modules, the class of generalized extending modules such that either for every submodule or a kind of special submodules, there exists a direct summand with the property that the direct sum of the after mentioned submodules with the direct summand is essentially contained in the module. The authors introduce various generalization of extending modules of this type, including weak CS-modules; \(C_{11}\)-modules, \(C_{11}\)-rings, and introduce a framework which encompasses most of the generalizations of the CS property and allow them to target specific sets of submodules of a module for application of the CS property. The fifth chapter is devoting to the study on the so-called outer generalization of extending modules, the class of generalized extending modules, which are based on a technical condition like a homomorphism into a direct summand or an equivalence relation in the lattice of submodules, etc. The authors introduce and study various generalization of extending modules, including \(C_{12}\)-modules, GLS-modules, G-extending modules, and so on. Chapter 6 studies the dual Goldie and ec-complement versions of the extending property. Chapter 7 formulates open problems and questions. The problems are legitimate and provide a basis for further research which in turn will greatly broaden the scope of the theory. The appendix of the book contains a construction of rings of quotients via Gabriel topologies. The book ends with a quite complete bibliography.

The book is addressed to students and researchers in module and ring theory. It will undoubtedly be a valuable tool for everybody working in this area. Finally, it should be mentioned that the authors have made a well-thought-out effort to improve and refine the terminology on the subject.

Reviewer: Bin Zhu (Beijing)

### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16D10 | General module theory in associative algebras |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16D50 | Injective modules, self-injective associative rings |

16D40 | Free, projective, and flat modules and ideals in associative algebras |