## Almost Free Modules: Set-Theoretic MethodsThis is an extended treatment of the set-theoretic techniques which have transformed the study of abelian group and module theory over the last 15 years. Part of the book is new work which does not appear elsewhere in any form. In addition, a large body of material which has appeared previously (in scattered and sometimes inaccessible journal articles) has been extensively reworked and in many cases given new and improved proofs. The set theory required is carefully developed with algebraists in mind, and the independence results are derived from explicitly stated axioms. The book contains exercises and a guide to the literature and is suitable for use in graduate courses or seminars, as well as being of interest to researchers in algebra and logic. |

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### Contents

1 | |

16 | |

CHAPTER III SLENDER MODULES | 51 |

CHAPTER IV ALMOST FREE MODULES | 82 |

CHAPTER V PUREINJECTIVE MODULES | 118 |

CHAPTER VI MORE SET THEORY | 138 |

CHAPTER VII ALMOST FREE MODULES REVISITED IV VI | 180 |

CHAPTER VIII COUNTABLYSEPARABLE GROUPS VI VIIl | 247 |

CHAPTER X ITERATED SUMS AND PRODUCTS III | 300 |

CHAPTER XI TOPOLOGICAL METHODS X IV | 313 |

CHAPTER XII THE STRUCTURE OF EXT VIIVIIIl | 344 |

CHAPTER XIII THE BLACK BOX AND ENDOMORPHISM RINGS V VI | 389 |

CHAPTER XIV DUAL GROUPS IX XI XIII | 420 |

OPEN PROBLEMS | 453 |

456 | |

477 | |

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### Common terms and phrases

abelian group assume Axiom belongs cardinality N1 choose clopen cofinality consistent construction contained contradiction Corollary cotorsion countable sets define definition denote direct sum direct summand disjoint dual group element endomorphism Exercise exists Ext(A filter finite subset free group free module function group of cardinality hence Hint holds homomorphism hypothesis II;er implies increasing sequence induction infinite isomorphic k-complete k-filtration K-generated ladder system Lemma Let F limit ordinal measurable cardinal n e w N1-free N1-separable group non-free non-reflecting non-zero p-adic predual projective PROOF Proposition prove pure closure pure subgroup pure-injective R-module rank reflexive regular cardinal Reid class ring satisfies short exact sequence slender stationary set stationary subset strongly k-free submodule Suppose surjective Theorem topology torsion-free torsionless ultrafilter uncountable cardinal W-group w-measurable weakly compact X-system Zºº