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Content Algebras and Zero-Divisors

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https://osnadocs.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-201102107989
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dc.contributor.advisorProf. Dr. Winfried Bruns
dc.creatorNasehpour, Peyman
dc.date.accessioned2011-02-10T13:00:32Z
dc.date.available2011-02-10T13:00:32Z
dc.date.issued2011-02-10T13:00:32Z
dc.identifier.urihttps://osnadocs.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-201102107989-
dc.description.abstractThis thesis concerns two topics. The first topic, that is related to the Dedekind-Mertens Lemma, the notion of the so-called content algebra, is discussed in chapter 2. Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = \cap \lbrace I \colon I \text{~is an ideal of~} R \text{~and~} x \in IM \rbrace $. $M$ is said to be a \textit{content} $R$-module if $x \in c(x)M $, for all $x \in M$. The $R$-algebra $B$ is called a \textit{content} $R$-algebra, if it is a faithfully flat and content $R$-module and it satisfies the Dedekind-Mertens content formula. In chapter 2, it is proved that in content extensions, minimal primes extend to minimal primes, and zero-divisors of a content algebra over a ring which has Property (A) or whose set of zero-divisors is a finite union of prime ideals are discussed. The preservation of diameter of zero-divisor graph under content extensions is also examined. Gaussian and Armendariz algebras and localization of content algebras at the multiplicatively closed set $S^ \prime = \lbrace f \in B \colon c(f) = R \rbrace$ are considered as well. In chapter 3, the second topic of the thesis, that is about the grade of the zero-divisor modules, is discussed. Let $R$ be a commutative ring, $I$ a finitely generated ideal of $R$, and $M$ a zero-divisor $R$-module. It is shown that the $M$-grade of $I$ defined by the Koszul complex is consistent with the definition of $M$-grade of $I$ defined by the length of maximal $M$-sequences in I$. Chapter 1 is a preliminarily chapter and dedicated to the introduction of content modules and also locally Nakayama modules.eng
dc.rightsNamensnennung-NichtKommerziell-KeineBearbeitung 3.0 Unported-
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/-
dc.subjectcontent moduleeng
dc.subjectcontent algebraeng
dc.subjectlocally Nakayama moduleeng
dc.subjectweak content algebraeng
dc.subjectfew zero-divisorseng
dc.subjectGaussian algebraeng
dc.subjectArmendariz algebraeng
dc.subjectzero-divisor grapheng
dc.subjectGradeeng
dc.subjecthomological dimensioneng
dc.subjectProperty (A)eng
dc.subjectZero-divisor moduleeng
dc.subjectsemigroup ringeng
dc.subjectsemigroup moduleeng
dc.subjectlocal cohomological moduleeng
dc.subjectMcCoy's propertyeng
dc.subjectminimal primeeng
dc.subjectprimal ringeng
dc.subject.ddc510 - Mathematik
dc.titleContent Algebras and Zero-Divisorseng
dc.title.alternativeInhaltsalgebren und Nullteilerger
dc.typeDissertation oder Habilitation [doctoralThesis]-
thesis.locationOsnabrück-
thesis.institutionUniversität-
thesis.typeDissertation [thesis.doctoral]-
thesis.date2011-01-27-
dc.contributor.refereeProf. Siamak Yassemi
dc.subject.bk31.23 - Ideale, Ringe, Moduln, Algebren
dc.subject.bk31.12 - Kombinatorik, Graphentheorie
dc.subject.msc13-02 - Research exposition
vCard.ORGFB6
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