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**Lifting modules. Supplements and projectivity in module theory.**
*(English)*
Zbl 1102.16001

Frontiers in Mathematics. Basel: Birkhäuser (ISBN 3-7643-7572-8/pbk). xii, 394 p. (2006).

An \(R\)-module is lifting if every submodule \(N\) of \(M\) contains a direct summand \(X\) of \(M\) such that \(N/X\) is small in \(M/X\). This concept is a generalization of projectivity and is the dual of extending modules. The objective of this book is to produce a reference work with respect to lifting modules on par with that existing for extending modules [N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending modules (Pitman Res. Notes Math. Ser. 313, Longman Scientific and Technical, Harlow) (1994; Zbl 0841.16001)]. This effort entails much more than just a straight forward dualization of results on extending modules, since complements (in terms of which extending modules are defined) exist for any submodule of a module, whereas supplements of a submodule need not exist.

The first four chapters pave the way for the study of lifting modules in Chapter 5. Since the existence of supplements in a module is not a direct consequence of (weak) projectivity conditions, it is often necessary to rely on either finiteness or structural conditions on the lattice of submodules of \(M\); hence the need for the extensive preparatory Chapters 1 through 4.

Basic notions concerning injectivity and in particular, projectivity are discussed in Chapter 1. Chapter 2 highlights torsion-theoretic aspects such as preradicals, colocalisation, torsion theories associated with small submodules and projectivity, and proper classes.

Decompositions or indecomposability of modules play a vital role in the study of lifting modules (as in much of the study of module theory). The relevant results are discussed in Chapter 3, including exchange properties, LE-modules, local direct summands, total and LE-decompositions and stable range.

The key concept of supplements is studied in Chapter 4. The class of semilocal modules, and semilocal rings, are studied in the first part of the chapter. Weak supplements are introduced and it is shown that they do not always exist but a connection between hollow dimension and the existence of weak supplements is found. Different classes of supplemented modules are introduced and studied.

Finally, in Chapter 5, lifting modules themselves are introduced and studied. The indecomposable lifting modules are shown to be exactly the hollow modules and any lifting module of finite uniform or finite hollow dimension are proven to be a finite direct sum of hollow modules. Several more conditions are found for a lifting module to be a direct sum of indecomposables. Conditions for a direct sum of lifting modules to be lifting are found.

Several generalizations and weaker lifting are also studied, and the relationships between them and decomposition and closure with respect to factor modules and direct sums are investigated.

Harada modules (every injective module is lifting) and co-Harada modules and their relationship with quasi-Frobenius modules are studied. Characterizations of modules in \(\sigma[M]\), the full subcategory of \(R\)-Mod generated by \(M\), for which all extending modules are lifting, and of modules for which all modules are lifting are found in the last paragraph.

The text is interspersed by numerous exercises and details of two graph theoretical techniques needed earlier in the book are given in an appendix.

The first four chapters pave the way for the study of lifting modules in Chapter 5. Since the existence of supplements in a module is not a direct consequence of (weak) projectivity conditions, it is often necessary to rely on either finiteness or structural conditions on the lattice of submodules of \(M\); hence the need for the extensive preparatory Chapters 1 through 4.

Basic notions concerning injectivity and in particular, projectivity are discussed in Chapter 1. Chapter 2 highlights torsion-theoretic aspects such as preradicals, colocalisation, torsion theories associated with small submodules and projectivity, and proper classes.

Decompositions or indecomposability of modules play a vital role in the study of lifting modules (as in much of the study of module theory). The relevant results are discussed in Chapter 3, including exchange properties, LE-modules, local direct summands, total and LE-decompositions and stable range.

The key concept of supplements is studied in Chapter 4. The class of semilocal modules, and semilocal rings, are studied in the first part of the chapter. Weak supplements are introduced and it is shown that they do not always exist but a connection between hollow dimension and the existence of weak supplements is found. Different classes of supplemented modules are introduced and studied.

Finally, in Chapter 5, lifting modules themselves are introduced and studied. The indecomposable lifting modules are shown to be exactly the hollow modules and any lifting module of finite uniform or finite hollow dimension are proven to be a finite direct sum of hollow modules. Several more conditions are found for a lifting module to be a direct sum of indecomposables. Conditions for a direct sum of lifting modules to be lifting are found.

Several generalizations and weaker lifting are also studied, and the relationships between them and decomposition and closure with respect to factor modules and direct sums are investigated.

Harada modules (every injective module is lifting) and co-Harada modules and their relationship with quasi-Frobenius modules are studied. Characterizations of modules in \(\sigma[M]\), the full subcategory of \(R\)-Mod generated by \(M\), for which all extending modules are lifting, and of modules for which all modules are lifting are found in the last paragraph.

The text is interspersed by numerous exercises and details of two graph theoretical techniques needed earlier in the book are given in an appendix.

Reviewer: Frieda Theron (Pretoria)

### MSC:

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

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

16E10 | Homological dimension in associative algebras |