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Statements

Subject Item
n2:RIV%2F00216208%3A11320%2F14%3A10287247%21RIV15-MSM-11320___
rdf:type
skos:Concept n18:Vysledek
rdfs:seeAlso
http://dx.doi.org/10.1016/j.jalgebra.2013.09.031
dcterms: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.
dcterms:title
Weakly based modules over Dedekind domains Weakly based modules over Dedekind domains
skos:prefLabel
Weakly based modules over Dedekind domains Weakly based modules over Dedekind domains
skos:notation
RIV/00216208:11320/14:10287247!RIV15-MSM-11320___
n4:aktivita
n10:I n10:P
n4:aktivity
I, P(GA201/09/0816)
n4:cisloPeriodika
399
n4:dodaniDat
n12:2015
n4:domaciTvurceVysledku
n17:1806068 n17:8174121
n4:druhVysledku
n9:J
n4:duvernostUdaju
n6:S
n4:entitaPredkladatele
n8:predkladatel
n4:idSjednocenehoVysledku
56018
n4:idVysledku
RIV/00216208:11320/14:10287247
n4:jazykVysledku
n16:eng
n4:klicovaSlova
Semisimple; Torsion; Local; Dedekind domain; Weak basis; Minimal generating set
n4:klicoveSlovo
n5:Local n5:Dedekind%20domain n5:Weak%20basis n5:Minimal%20generating%20set n5:Torsion n5:Semisimple
n4:kodStatuVydavatele
US - Spojené státy americké
n4:kontrolniKodProRIV
[981304877002]
n4:nazevZdroje
Journal of Algebra
n4:obor
n19:BA
n4:pocetDomacichTvurcuVysledku
2
n4:pocetTvurcuVysledku
2
n4:projekt
n11:GA201%2F09%2F0816
n4:rokUplatneniVysledku
n12:2014
n4:svazekPeriodika
2014
n4:tvurceVysledku
Hrbek, Michal Růžička, Pavel
n4:wos
000330006200013
s:issn
0021-8693
s:numberOfPages
18
n15:doi
10.1016/j.jalgebra.2013.09.031
n3:organizacniJednotka
11320