##
**Extending modules.**
*(English)*
Zbl 0841.16001

Pitman Research Notes in Mathematics Series. 313. Harlow: Longman Scientific & Technical. xvii, 224 p. (1994).

The theory of extending modules, together with that of continuous modules and other generalisations of injective modules can be regarded as being part of a tradition which goes back to von Neumann’s work on continuous geometries. On the other hand, without knowing this history one can think of extending modules as giving a natural simultaneous generalisation of injective modules and of semi-simple modules, and as such they present a natural area for study in general module theory. Over recent years a lot of work has been done in this and related areas, much of it by the authors of this book. They have rightly felt that the time has come to collect this material together, to give it a unified treatment, to survey the relevant work, and to state open problems. This work will be an invaluable source of information for those working in the area.

Let \(R\) be any associative ring with identity element, and let \(M\) be a unital right \(R\)-module. Then \(M\) is said to be an extending module (sometimes known as a CS module) if every submodule of \(M\) is an essential submodule of a direct summand of \(M\). Injective modules, semi-simple modules, and uniform modules are easily seen to be examples of extending modules. But the direct sum of two extending modules is in general not extending, even over the ring of integers. This suggests that the theory of extending modules is likely to be interestingly non-straightforward. It raises questions such as: For which rings \(R\) is every right \(R\)-module extending? When is the direct sum of two extending modules extending? The present book considers many questions of this sort about extending modules, and uses the theory of extending modules to give a unified approach to proving results about other types of modules and rings.

One theme of the work reported here is to fix an \(R\)-module \(M\) and to consider only those \(R\)-modules which are influenced by \(M\) (in the sense of being submodules of factor modules of direct sums of copies of \(M\)), or more restrictively only those modules which are submodules of factor modules of \(M\). This has made it possible to extend certain results which were first proved for rings to apply to an arbitrary module \(M\) and retrieve the original result by taking \(M=R\). A good example of this is Osofsky’s theorem that if every cyclic right \(R\)-module is injective then \(R\) is semi-simple Artinian. It is now known that this and many other results follow from the Osofsky-Smith theorem: Let \(M\) be a cyclic right \(R\)-module such that every factor module of every cyclic submodule of \(M\) is extending; then \(M\) is a direct sum of uniform modules. This result in turn has undergone considerable extension. A similar fate (and vast improvement) has befallen the reviewer’s result that if every cyclic right \(R\)-module is the direct sum of a projective module and a Noetherian module then \(R\) is right Noetherian. This book is now the natural place to look for results which characterise a ring by properties of its cyclic or finitely-generated modules.

For much of the book the base-ring \(R\) is irrelevant and the results are purely module-theoretic. For instance: A finitely-generated extending module which satisfies the ascending (descending) chain condition for essential submodules is Noetherian (Artinian). There are also ring-theoretic results such as numerous characterisations of QF-rings, and in 13.5 several answers are given to the question as to which rings are characterised by the property that all their modules are extending (one answer is that they are Artinian serial rings with square-zero radical).

The presentation is self-contained to the extent that definitions are given and basic properties are summarised, but the reader who wishes to gain a thorough understanding of why things turn out as they do will have to do some background reading, in particular of R. Wisbauer’s book [Foundations of module and ring theory (1991; Zbl 0746.16001)]. The results are stated in such a way that they can be understood once the definitions have been grasped; in this connection the authors are to be congratulated for not using the cryptic abbreviations which have made some of the papers in this area very hard to read. This book will be a very useful source of information for those interested in generalizations of injective modules, characterisations of rings by properties of their modules, and many other questions in module theory.

Let \(R\) be any associative ring with identity element, and let \(M\) be a unital right \(R\)-module. Then \(M\) is said to be an extending module (sometimes known as a CS module) if every submodule of \(M\) is an essential submodule of a direct summand of \(M\). Injective modules, semi-simple modules, and uniform modules are easily seen to be examples of extending modules. But the direct sum of two extending modules is in general not extending, even over the ring of integers. This suggests that the theory of extending modules is likely to be interestingly non-straightforward. It raises questions such as: For which rings \(R\) is every right \(R\)-module extending? When is the direct sum of two extending modules extending? The present book considers many questions of this sort about extending modules, and uses the theory of extending modules to give a unified approach to proving results about other types of modules and rings.

One theme of the work reported here is to fix an \(R\)-module \(M\) and to consider only those \(R\)-modules which are influenced by \(M\) (in the sense of being submodules of factor modules of direct sums of copies of \(M\)), or more restrictively only those modules which are submodules of factor modules of \(M\). This has made it possible to extend certain results which were first proved for rings to apply to an arbitrary module \(M\) and retrieve the original result by taking \(M=R\). A good example of this is Osofsky’s theorem that if every cyclic right \(R\)-module is injective then \(R\) is semi-simple Artinian. It is now known that this and many other results follow from the Osofsky-Smith theorem: Let \(M\) be a cyclic right \(R\)-module such that every factor module of every cyclic submodule of \(M\) is extending; then \(M\) is a direct sum of uniform modules. This result in turn has undergone considerable extension. A similar fate (and vast improvement) has befallen the reviewer’s result that if every cyclic right \(R\)-module is the direct sum of a projective module and a Noetherian module then \(R\) is right Noetherian. This book is now the natural place to look for results which characterise a ring by properties of its cyclic or finitely-generated modules.

For much of the book the base-ring \(R\) is irrelevant and the results are purely module-theoretic. For instance: A finitely-generated extending module which satisfies the ascending (descending) chain condition for essential submodules is Noetherian (Artinian). There are also ring-theoretic results such as numerous characterisations of QF-rings, and in 13.5 several answers are given to the question as to which rings are characterised by the property that all their modules are extending (one answer is that they are Artinian serial rings with square-zero radical).

The presentation is self-contained to the extent that definitions are given and basic properties are summarised, but the reader who wishes to gain a thorough understanding of why things turn out as they do will have to do some background reading, in particular of R. Wisbauer’s book [Foundations of module and ring theory (1991; Zbl 0746.16001)]. The results are stated in such a way that they can be understood once the definitions have been grasped; in this connection the authors are to be congratulated for not using the cryptic abbreviations which have made some of the papers in this area very hard to read. This book will be a very useful source of information for those interested in generalizations of injective modules, characterisations of rings by properties of their modules, and many other questions in module theory.

Reviewer: A.W.Chatters (Bristol)

### MSC:

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

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

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

16D90 | Module categories in associative algebras |

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

16P40 | Noetherian rings and modules (associative rings and algebras) |