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Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 58). Available also in a digital version from Dissertation Abstracts.

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This thesis investigates some of the properties of R-modules. The material is presented in three chapters. Definitions and theorems which are assumed are stated in Chapter I. Proofs of these theorems may be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1958. It is assumed that the reader is familiar with the basic properties of commutative rings and ideals in rings. Properties of R-modules are developed in Chapter II. The most important results presented in this chapter include existence theorems for R-modules and properties of submodules in R-modules. The third and final chapter presents an example which illustrates how a ring R, may be regarded as an R-module and speaks of the direct sum of ideals of a ring as a direct sum of submodules.

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This thesis investigates some of the properties of commutative rings which do not necessarily contain a multiplicative identity (unity).

This thesis investigates some of the properties of valuation rings. It is assumed that the reader is familiar with the basic properties of commutative rings and ideals in rings. Unless otherwise stated, all rings considered in this thesis are commutative rings with a unity.

This thesis is a study of some properties of prime ideals in commutative rings with unity.

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.