Abstract: This thesis concerns two topics. The first topic, that is related to the DedekindMertens
Lemma, the notion of the socalled 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 DedekindMertens content formula. In chapter 2, it is proved that in content extensions, minimal
primes extend to minimal primes, and zerodivisors of a content algebra over a ring which has Property (A) or whose set of
zerodivisors is a finite union of prime ideals are discussed. The preservation of diameter of zerodivisor 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 zerodivisor modules, is discussed. Let
$R$ be a commutative ring, $I$ a finitely generated ideal of $R$, and $M$ a zerodivisor $R$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.
Supervisor: Prof. Dr. Winfried Bruns
Important keywords and phrases: modules, commutative rings and algebras, content module, content algebra,
weak content algebra, very few zerodivisor, zerodivisor graph, Gaussian algebra, Armendariz algebra, minimal prime ideal,
property (A), Gauss' lemma, DedekindMertens lemma, ZDmodule, grade, local cohomology module, homological dimension, McCoy's
property, semigroup ring, semigroup module, locally Nakayama module.
