Lattice Concepts of Module Theory

Grigore Calugareanu (Autor)

Buch | Softcover

225 Seiten

2010 | Softcover reprint of hardcover 1st ed. 2000
Springer (Verlag)
978-90-481-5530-9 (ISBN)

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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.

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.

Erscheint lt. Verlag	8.12.2010
Reihe/Serie	Texts in the Mathematical Sciences ; 22
Zusatzinfo	XIII, 225 p.
Verlagsort	Dordrecht
Sprache	englisch
Maße	170 x 244 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
ISBN-10	90-481-5530-4 / 9048155304
ISBN-13	978-90-481-5530-9 / 9789048155309
Zustand	Neuware