About: Weakly based modules over Dedekind domains

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
rdfs:seeAlso	http://dx.doi.org/10.1016/j.jalgebra.2013.09.031
Description	We say that a subset X of a left R-module M is weakly independent provided that whenever a(1)x(1) + ... + a(n)x(n) = 0 for pairwise distinct elements x(1), ... , x(n) form X, then none of a(1), ... , a(n) is invertible in R. Weakly independent generating sets (we call them weak bases) are exactly generating sets minimal with respect to inclusion. The aim of the paper is to characterize modules over Dedekind domains possessing a weak basis. We will characterize them as follows: Let R be a Dedekind domain and let M be a x-generated R-module, for some infinite cardinal x. Then M has a weak basis iff at least one of the following conditions is satisfied: (1) There are two different prime ideals P, Q of R such that dim(R/P) (M/PM) = dim(R/Q) (M/QM) = x; (2) There are a prime ideal P of R and a decomposition M similar or equal to F circle plus N where F is a free module and dim(R/P) (tau N/P tau N) = gen(N); (3) There is a projection of M onto an R-module circle plus(P is an element of Spec(R)) V-P, where V-P is a vector space over R/P with dim(R/P)(V-P) < x for each P is an element of Spec(R) and Sigma(P is an element of Spec(R)) dim(R/P)(V-P)=x. (C) 2013 Elsevier Inc. All rights reserved. We say that a subset X of a left R-module M is weakly independent provided that whenever a(1)x(1) + ... + a(n)x(n) = 0 for pairwise distinct elements x(1), ... , x(n) form X, then none of a(1), ... , a(n) is invertible in R. Weakly independent generating sets (we call them weak bases) are exactly generating sets minimal with respect to inclusion. The aim of the paper is to characterize modules over Dedekind domains possessing a weak basis. We will characterize them as follows: Let R be a Dedekind domain and let M be a x-generated R-module, for some infinite cardinal x. Then M has a weak basis iff at least one of the following conditions is satisfied: (1) There are two different prime ideals P, Q of R such that dim(R/P) (M/PM) = dim(R/Q) (M/QM) = x; (2) There are a prime ideal P of R and a decomposition M similar or equal to F circle plus N where F is a free module and dim(R/P) (tau N/P tau N) = gen(N); (3) There is a projection of M onto an R-module circle plus(P is an element of Spec(R)) V-P, where V-P is a vector space over R/P with dim(R/P)(V-P) < x for each P is an element of Spec(R) and Sigma(P is an element of Spec(R)) dim(R/P)(V-P)=x. (C) 2013 Elsevier Inc. All rights reserved. (en)
Title	Weakly based modules over Dedekind domains Weakly based modules over Dedekind domains (en)
skos:prefLabel	Weakly based modules over Dedekind domains Weakly based modules over Dedekind domains (en)
skos:notation	RIV/00216208:11320/14:10287247!RIV15-MSM-11320___
http://linked.open...avai/riv/aktivita	I P
http://linked.open...avai/riv/aktivity	I, P(GA201/09/0816)
http://linked.open...iv/cisloPeriodika	399
http://linked.open...vai/riv/dodaniDat	2015
http://linked.open...aciTvurceVysledku	Růžička, Pavel Hrbek, Michal
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	56018
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/14:10287247
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	Semisimple; Torsion; Local; Dedekind domain; Weak basis; Minimal generating set (en)
http://linked.open.../riv/klicoveSlovo	Torsion Local Dedekind domain Minimal generating set Semisimple Weak basis
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[981304877002]
http://linked.open...i/riv/nazevZdroje	Journal of Algebra
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	2 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...vavai/riv/projekt	Algebraické metody teorie reprezentací (aproximace, realizace a omezení)
http://linked.open...UplatneniVysledku	2014
http://linked.open...v/svazekPeriodika	2014
http://linked.open...iv/tvurceVysledku	Hrbek, Michal Růžička, Pavel
http://linked.open...ain/vavai/riv/wos	000330006200013
issn	0021-8693
number of pages	18 (xsd:int)
http://bibframe.org/vocab/doi	10.1016/j.jalgebra.2013.09.031
http://localhost/t...ganizacniJednotka	11320

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