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＜電子ブック＞
Lattice Concepts of Module Theory / by Grigore Calugareanu
(Texts in the Mathematical Sciences ; 22)

版	1st ed. 2000.
出版者	(Dordrecht : Springer Netherlands : Imprint: Springer)
出版年	2000
本文言語	英語
大きさ	XIII, 225 p : online resource
著者標目	*Calugareanu, Grigore author SpringerLink (Online service)
件　名	LCSH:Algebra LCSH:Group theory LCSH:Associative rings LCSH:Associative algebras LCSH:Commutative algebra LCSH:Commutative rings FREE:Order, Lattices, Ordered Algebraic Structures FREE:Group Theory and Generalizations FREE:Associative Rings and Algebras FREE:Commutative Rings and Algebras FREE:Algebra
一般注記	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 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 HTTP:URL=https://doi.org/10.1007/978-94-015-9588-9

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