In this thesis we describe how the balancing of the $\operatorname{Tor}$ functor can be used to compute the minimal free resolution of a graded module $M$ over the polynomial ring $B=\mathbb{K}[X_0,\dots,X_m]$ ($\mathbb{K}$ a field $X_i$'s indeterminates). Using a correspondence due to R. Stanley and M. Hochster, we explicitly show how this approach can be used in the case when $M=\mathbb{K}[S]$, the semigroup ring of a subsemigroup $S\subseteq \mathbb{N}^l$ (containing $0$) over $\mathbb{K}$ and when $M$ is a monomial ideal of $B$. We also study the class of affine semigroup rings for which $\mathbb{K}[S]\cong B/\mathfrak{p}$ is the homogeneous coordinate ring of a monomial curve in $\mathbb{P}^n_{\mathbb{K}}$. We use easily computable combinatorial and arithmetic properties of $S$ to define a notion which we call stabilization. We provide a direct proof showing how stabilization gives a bound on the $\mathbb{N}$-graded degree of minimal generators of $\mathfrak{p}$ and also show that it is related to the regularity of $\mathfrak{p}$. Moreover, we partition the above mentioned class into three cases and show that this partitioning is reflected in how the regularity is attained. An interesting consequence is that the regularity of $\mathfrak{p}$ can be effectively computed by elementary means. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2008-09-24 09:49:35.462

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Jan. 13, 2009). PDF text: 40 p. ; 747 K. UMI publication number: AAT 3315330. Includes bibliographical references. Also available in microfilm and microfiche formats.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1963. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

This thesis exhibits a collection of proofs of theorems on ideals in a commutative ring with and without a unity. Theorems treated involve properties of ideals under certain operations (sum, product, quotient, intersection, and union); properties of homomorphic mappings of ideals; contraction and extension theorems concerning ideals and quotient rings of domains with respect to multiplicative systems; properties of maximal, minimal, prime, semi-prime, and primary ideals; properties of radicals of ideals with relations to quotient rings, semi-prime, and primary ideals.

In this thesis, we are concerned with various aspects of fuzzy ideals of commutative rings. The central theorem is that of primary decomposition of a fuzzy ideal as an intersection of fuzzy primary ideals in a commutative Noetherian ring. We establish the existence and the two uniqueness theorems of primary decomposition of any fuzzy ideal with membership value 1 at the zero element. In proving this central result, we build up the necessary tools such as fuzzy primary ideals and the related concept of fuzzy maximal ideals, fuzzy prime ideals and fuzzy radicals. Another approach explores various characterizations of fuzzy ideals, namely, generation and level cuts of fuzzy ideals, relation between fuzzy ideals, congruences and quotient fuzzy rings. We also tie up several authors' seemingly different definitions of fuzzy prime, primary, semiprimary and fuzzy radicals available in the literature and show some of their equivalences and implications, providing counter-examples where certain implications fail.

M.Sc. / This thesis is concerned with one possible interplay between commutative algebra and graph theory. Specifically, we associate with a commutative ring R a graph and then set out to determine how the ring's properties influence the chromatic and clique numbers of the graph. The graph referred to is obtained by letting each ring element be represented by a vertex in the graph and joining two vertices when the product of their corresponding ring elements is equal to zero. The thesis focuses on rings that have a finite chromatic number, where the chromatic number of the ring is equal to the chromatic number of the associated graph. The nilradical of the ring plays a prominent role in these- investigations. Furthermore, the thesis also discusses conditions under which the chromatic and clique numbers of the associated graph are equal. The thesis ends with a discussion of rings with low (< 5) chromatic number and an example of a ring with clique number 5 and chromatic number 6.

No description available.

This paper is an investigation of several basic properties of ordered Abelian groups, valuations, the relationship between valuation rings, valuations, and their value groups and valuation rings. The proofs to all theorems stated without proof can be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1858. In Chapter I several basic theorems which are used in later proofs are stated without proof, and we prove several theorems on the structure of ordered Abelian groups, and the basic relationships between these groups, valuations, and their valuation rings in a field. In Chapter II we deal with valuation rings, and relate the structure of valuation rings to the structure of their value groups.

by Chen Yuqun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references and index. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

This dissertation focuses on the properties of a domain which has the property that each ideal is a finite intersection of a π-ideal, the properties of a domain which have the property that each ideal is a finite product of π-ideal, and the containment relations of the resulting classes of ideals. Chapter 1 states definitions which are needed in later chapters. Chapters 2 and 3 focuses on domains which have the property that each ideal in D is a finite intersection of π-ideals while Chapter 4 focuses on domains with the property that each ideal is a finite product of π-ideals. Chapter 5 discusses the containment relations which occur as a result of Chapters 2 and 3.