Lattice Concepts of Module Theory / Edition 1

by Grigore Calugareanu

Overview

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 structure 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 concepts (e. g.

Product Details

ISBN-13:	9780792364887
Publisher:	Springer Netherlands
Publication date:	08/31/2000
Series:	Texts in the Mathematical Sciences , #22
Edition description:	2000
Pages:	225
Product dimensions:	6.14(w) x 9.21(h) x 0.36(d)

Table of Contents

1 Basic notions and results.- 2 Compactly generated lattices.- 3 Composition series. Decompositions.- 4 Essential elements. Pseudo—complements.- 5 Socle. Torsion lattices.- 6 Independence. Semiatomic lattices.- 7 Radical. Superfluous and fully invariant elements.- 8 Lattices of finite uniform dimension.- 9 Purity and neatness in lattices.- 10 Coatomic lattices.- 11 Co—compact lattices.- 12 Supplemented lattices. Locally artinian lattices.- 13 Several dimensions.- 14 Solutions of exercises.

Editorial Reviews

Presents all module-theoretic results that can be proved by lattice theory only, and develops the theory necessary to do so. Results treated fall into categories such as the origins of lattice theory, module-theoretic results generalized in modular and compactly generated lattices, very special module-theoretic results generalized in lattices, and new concepts in lattices introduced by the author. Contains chapters on pseudo-complements, torsion lattices, semiatomic lattices, lattices of finite uniform dimension, supplemented lattices, and coatomic lattices. The author teaches algebra at Babes-Bolyai University, Romania. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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