Anderson and Frazier defined a generalization of factorization in integral domains called tau-factorization. If D is an integral domain and tau is a symmetric relation on the nonzero nonunits of D, then a tau-factorization of a nonzero nonunit a in D is an expression a = lambda a_1 ... a_n, where lambda is a unit in D, each a_i is a nonzero nonunit in D, and a_i tau a_j for i != j. If tau = D^# x D^#, where D^# denotes the nonzero nonunits of D, then the tau-factorizations are just the usual factorizations, and with other choices of tau we get interesting variants on standard factorization. For example, if we define a tau_d b if and only if (a, b) = D, then the tau_d-factorizations are the comaximal factorizations introduced by McAdam and Swan. Anderson and Frazier defined tau-factorization analogues of many different factorization concepts and properties, and proved a number of theorems either generalizing standard factorization results or the comaximal factorization results of McAdam and Swan. Some of these concepts include tau-UFD's, tau-atomic domains, the tau-ACCP property, tau-BFD's, tau-FFD's, and tau-HFD's. They showed the implications between these concepts and showed how each of the standard variations implied their tau-factorization counterparts (sometimes assuming certain natural constraints on tau). Later, Ortiz-Albino introduced a new concept called Gamma-factorization that generalized tau-factorization. We will summarize the known theory of tau-factorization and Gamma-factorization as well as introduce several new or improved results.

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 work focuses on commutative algebra, its combinatorial and computational aspects, and its interactions with statistics. The main objects of interest are projective varieties in Pn, algebraic properties of their coordinate rings, and the combinatorial invariants, such as Hilbert series and Gröbner fans, of their defining ideals. Specifically, the ideals in this work are all toric ideals, and they come in three flavors: they are defining ideals of a family of classical varieties called rational normal scrolls, cut ideals that can be associated to a graph, and phylogenetic ideals arising in a new and increasingly popular area of algebraic statistics.

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We consider and solve some open conjectures on the asymptotic behavior of the number of different numbers of the nonassociate factorizations of prescribed minimal length for specific finite factorization domains. The asymptotic behavior will be classified for Cohen-Kaplansky domains in Chapter 1 and for domains of the form R=K+XF[X] for finite fields K and F in Chapter 2. A corollary of the main result in Chapter 3 will determine the asymptotic behavior for Krull domains with finite divisor class group.

Thesis (Ph.D.)--University of Delaware, 2010. / Principal faculty advisor: Robert Coulter, Dept. of Mathematical Sciences. Includes bibliographical references.

We use the behavioural approach and commutative algebra to define and characterize poles and zeros of multidimensional (nD) linear systems. In the case of a system with a standard input output structure we provide new definitions and characterizations of system, controllable and uncontrollable zeros and demonstrate strong relationships between the controllable poles and zeros and properties of the system transfer matrix, and we show that the uncontrollable zeros are in fact uncontrollable poles. We also show that we can regard the zero as a form of pole with respect to an additional form of input output structure imposed on the zero output sub-behaviour. In the case when the behaviour has a latent variable description we make a further distinction of the zeros into several other classes including observable, unobservable and invariant zeros. In addition we also introduce their corresponding controllable and uncontrollable zeros, such as the observable controllable, unobservable controllable, invariant controllable, observable uncontrollable, unobservable uncontrollable and invariant uncontrollable etc. We again demonstrate strong relationships between these and other types of zeros and provide physical interpretations in terms of exponential and polynomial exponential trajectories. In the 1D case of a state-space model we show that the definitions and characterizations of the observable controllable and invariant zeros correspond to the transmission zeros and the invariant zeros in the classical 1D framework. This then completes the correspondences between the behavioural definitions of poles and zeros and those classical poles and zeros which have an interpretation in nD.

In this thesis, we introduce the theory of locally nilpotent derivations and use it to compute certain ring invariants. We prove some results about quasi-extensions of derivations and use them to show that certain rings are non-rigid. Our main result states that if k is a field of characteristic zero, C is an affine k-domain and B = C[T,Y] / < T^nY - f(T) >, where n >= 2 and f(T) \in C[T] is such that delta^2(f(0)) != 0 for all nonzero locally nilpotent derivations delta of C, then ML(B) != k. This shows in particular that the ring B is not a polynomial ring over k.