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.

In dieser Arbeit beweisen wir für Optimierungsprobleme in L0-Moduln relevante Resultate und untersuchen Anwendungen für die Darstellung von Präferenzen. Im ersten Kapitel geht es um quasikonkave, monotone und lokale Funktionen von einem L0-Modul X nach L0, die wir robust darstellen. Im zweiten Kapitel entwickeln wir das Ekeland’sche Variationsprinzip für L0-Moduln, die eine L0-Metrik besitzen. Wir beweisen eine L0 -Variante einer Verallgemeinerung des Ekeland’schen Theorems. Der Beweis des Brouwerschen Fixpunktsatzes für Funktionen, die auf (L0)^d definiert sind, wird in Kapitel 3 behandelt. Wir definieren das Konzept des Simplexes in (L0)^d und beweisen, dass jede lokale, folgenstetige Funktion darauf einen Fixpunkt besitzt. Dies nutzen wir, um den Fixpunktsatz auch für Funktionen auf beliebigen abgeschlossenen, L0 -konvexen Mengen zu zeigen. Eine allgemeinere Struktur als L0 ist die bedingte Menge. Im vierten Kapitel behandeln wir bedingte topologische Vektorräume. Wir führen das Konzept der Dualität für bedingte Mengen ein und beweisen Theoreme der Funktionalanalysis darauf, unter anderem das Theorem von Banach-Alaoglu und Krein-Šmulian. Im fünften Kapitel widmen wir uns der Darstellung mit wandernden konvexen Mengen. Wir zeigen danach, wie die Transitivität für diese Darstellungsform beschrieben werden kann. Abschließend modellieren wir die Eigenschaft, dass die Transitivität einer Relation nur für ähnliche Elemente gesichert ist und diskutieren Arten der Darstellung solcher Relationen. / In this thesis, we prove results relevant for optimization problems in L0-modules and study applications to order theory. The first part deals with the notion of an Assessment Index (AI). For an L0 -module X an AI is a quasiconcave, monotone and local function mapping to L0. We prove a robust representation of these AIs. In the second chapter of this thesis, we develop Ekeland’s variational principle for L0-modules allowing for an L0-metric. We prove an L0-Version of a generalization of Ekeland’s theorem. A further application of L0 -theory is examined in the third chapter of this thesis, namely an extension of the Brouwer fixed point theorem to functions on (L0)^d . We define a conditional simplex, which is a simplex with respect to L0 , and prove that every local, sequentially continuous function has a fixed point. We extend the fixed point theorem to arbitrary closed, L0-convex sets. A more general structure than L0 -modules is the concept of conditional sets. In the fourth chapter of the thesis, we study conditional topological vector spaces. We examine the concept of duality for conditional sets and prove results of functional analysis: among others, the Banach-Alaoglu and the Krein-Šmulian theorem. Any L0 -module being a conditional set allows to apply all results to L0 -theory. In the fifth chapter, we discuss the property of transitivity of relations and its connection to certain forms of representations. After a survey of common representations of preferences, we attend to relations induced by moving convex sets which are relations of the form that x is preferred to y if and only if x − y is in a convex set depending on y. We examine in which cases such a representation is transitive. Finally, we exhibit nontransitivity due to dissimilarity of the compared object and discuss representations for relations of that type.

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