## Classes of ModulesBecause traditional ring theory places restrictive hypotheses on all submodules of a module, its results apply only to small classes of already well understood examples. Often, modules with infinite Goldie dimension have finite-type dimension, making them amenable to use with type dimension, but not Goldie dimension. By working with natural classes and type submodules (TS), Classes of Modules develops the foundations and tools for the next generation of ring and module theory. It shows how to achieve positive results by placing restrictive hypotheses on a small subset of the complement submodules, Furthermore, it explains the existence of various direct sum decompositions merely as special cases of type direct sum decompositions. Carefully developing the foundations of the subject, the authors begin by providing background on the terminology and introducing the different module classes. The modules classes consist of torsion, torsion-free, s[M], natural, and prenatural. They expand the discussion by exploring advanced theorems and new classes, such as new chain conditions, TS-module theory, and the lattice of prenatural classes of right R-modules, which contains many of the previously used lattices of module classes. The book finishes with a study of the Boolean ideal lattice of a ring. Through the novel concepts presented, Classes of Modules provides a new, unexplored direction to take in ring and module theory. |

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### Contents

1 | |

Important Module Classes and Constructions | 7 |

Finiteness Conditions | 33 |

Type Theory of Modules Dimension | 71 |

Type Theory of Modules Decompositions | 107 |

Lattices of Module Classes | 149 |

205 | |

215 | |

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### Common terms and phrases

ACC on Hk(R ascending chain condition atomic modules atomic submodule Boolean lattice Boolean ring chain conditions closed under quotient complement closure complement submodule complete lattice contains contradiction COROLLARY cyclic module Dauns decomposition defined direct sum direct summand essential submodule exists extends finite type dimension following are equivalent following hold fully invariant Goldie torsion hence hereditary pretorsion class hereditary torsion class hereditary torsion theory Hk(N homomorphism hypothesis implies indecomposable injective hulls isomorphic Ker(f LEMMA M-injective module M-natural Mod-R natural class nonsingular modules nonzero module nonzero submodule pairwise Orthogonal pre-natural class proof PROPOSITION quasi-injective quotient modules R-module right Noetherian right R-modules ring satisfies t-ACC simple module square free submodule of type suffices to show sum of M-injective Suppose t-DCC THEOREM torsion free class TP(R TS-module type closure type direct sum type submodule type summand uniform modules unique Zhou