Content Algebras and Zero-Divisors

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Titel: Content Algebras and Zero-Divisors
Sonstige Titel: Inhaltsalgebren und Nullteiler
Autor(en): Nasehpour, Peyman
Erstgutachter: Prof. Dr. Winfried Bruns
Zweitgutachter: Prof. Siamak Yassemi
Zusammenfassung: This 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.
Schlagworte: content module; content algebra; locally Nakayama module; weak content algebra; few zero-divisors; Gaussian algebra; Armendariz algebra; zero-divisor graph; Grade; homological dimension; Property (A); Zero-divisor module; semigroup ring; semigroup module; local cohomological module; McCoy's property; minimal prime; primal ring
Erscheinungsdatum: 10-Feb-2011
Enthalten in den Sammlungen:FB06 - E-Dissertationen

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