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Lattice Concepts of Module Theory

  • Book
  • © 2000

Overview

Authors:
  1. Grigore Călugăreanu

    1. Department of Algebra, Faculty of Mathematics and Computer Sciences, Babeş-Bolyai University, Cluj-Napoca, Romania

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Part of the book series: Texts in the Mathematical Sciences (TMS, volume 22)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Basic notions and results

    • Grigore Călugăreanu
    Pages 1-16

  3. Compactly generated lattices

    • Grigore Călugăreanu
    Pages 17-27

  4. Composition series. Decompositions

    • Grigore Călugăreanu
    Pages 29-38

  5. Essential elements. Pseudo—complements

    • Grigore Călugăreanu
    Pages 39-45

  6. Socle. Torsion lattices

    • Grigore Călugăreanu
    Pages 47-58

  7. Independence. Semiatomic lattices

    • Grigore Călugăreanu
    Pages 59-75

  8. Radical. Superfluous and fully invariant elements

    • Grigore Călugăreanu
    Pages 77-86

  9. Lattices of finite uniform dimension

    • Grigore Călugăreanu
    Pages 87-98

  10. Purity and neatness in lattices

    • Grigore Călugăreanu
    Pages 99-114

  11. Coatomic lattices

    • Grigore Călugăreanu
    Pages 115-122

  12. Co—compact lattices

    • Grigore Călugăreanu
    Pages 123-129

  13. Supplemented lattices. Locally artinian lattices

    • Grigore Călugăreanu
    Pages 131-142

  14. Several dimensions

    • Grigore Călugăreanu
    Pages 143-163

  15. Solutions of exercises

    • Grigore Călugăreanu
    Pages 165-218

  16. Back Matter

    Pages 219-225
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Keywords

About this book

It became more and more usual, from, say, the 1970s, for each book on Module Theory, to point out and prove some (but in no more than 15 to 20 pages) generalizations to (mostly modular) lattices. This was justified by the nowadays widely accepted perception that the structure of a module over a ring is best understood in terms of the lattice struc­ ture of its submodule lattice. Citing Louis H. Rowen "this important example (the lattice of all the submodules of a module) is the raison d'etre for the study of lattice theory by ring theorists". Indeed, many module-theoretic results can be proved by using lattice theory alone. The purpose of this book is to collect and present all and only the results of this kind, although for this purpose one must develop some significant lattice theory. The results in this book are of the following categories: the folklore of Lattice Theory (to be found in each Lattice Theory book), module­ theoretic results generalized in (modular, and possibly compactly gen­ erated) lattices (to be found in some 6 to 7 books published in the last 20 years), very special module-theoretic results generalized in lattices (e. g. , purity in Chapter 9 and several dimensions in Chapter 13, to be found mostly in [27], respectively, [34] and [18]) and some new con­ cepts (e. g.

Authors and Affiliations

  • Department of Algebra, Faculty of Mathematics and Computer Sciences, Babeş-Bolyai University, Cluj-Napoca, Romania

    Grigore Călugăreanu

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