Show simple item record

dc.contributor.advisorOrtiz-Albino, Reyes M.
dc.contributor.authorCalderón-Gómez, José E.
dc.description.abstractThe theory of $\tau$-factorizations, also known as theory of generalized factorizations, was developed by Anderson and Frazier in 2006. It was the result of a generalization of the comaximal factorizations by McAdams and Swam, replacing the condition of being comaximals to being related on the set of nonzero nonunits elements in the integral domain. Denote $D$ as an integral domain, $U(D)$ as the set of units of $D$ and $D^\#$ as the set of elements nonzero nonunits of $D$. The authors considered symmetric relations defined over the nonzero nonunits elements. The usual theory of factorizations came to be a particular case, where the relation used is $\tau=D^\#\times D^\#$. An expression of the form $a=\lambda a_1\cdot\cdot \cdot a_n$, where $\lambda\in U(D)$ and $a_i\tau a_j$ for all $1\leq i\neq j\leq n$, is called a $\tau$-factorizarion of $a$. Each $a_i$ is called a $\tau$-factor of $a$ and $a$ is a $\tau$-product of $a_i$. Furthermore, it is possible to obtain particular cases, such as factorizations in irreducibles elements, primals, and others, by taking $\tau=S\times S$, where $S$ is the set of irreducible elements or primals respectively. This work studied the relation $\tau_R$, where $R\subseteq D\times E$, $D$ and $E$ are integral domains, and $\tau$ is defined on $D^\#$. The relation $\tau_R$ is defined as $x\tau_R y$, if and only if there exist $a,b\in D^\#$ such that $a\tau b$, $aRx$, and $bRy$. That is, $\tau_R$ is ``the image of $\tau$ with respect to the relation $R$''. The properties of $\tau_R$ that can be inherited from $\tau$ in $\tau_R$ are analyzed . It must be clarified that although the definition is given with respect to the image of a relation, most of the work is focused in different types of functions, such as one to one and surjectives functions, homomorphisms, and others. The principal objective is to provide a way to study $\tau$-factorizations and structural properties using the images of the functions.en_US
dc.rightsCC0 1.0 Universal*
dc.subjectDominios de integridaden_US
dc.subject.lcshFactorization (Mathematics)en_US
dc.subject.lcshIntegral domainsen_US
dc.titleImagen de los τ-Productosen_US
dc.rights.holder(c) 2019 José Emilio Calderónen_US
dc.contributor.committeeOcasio, Victor
dc.contributor.committeeDziobiak, Stan
dc.contributor.representativeIrizarry, Zollianne Mathematicsen_US
dc.contributor.collegeCollege of Arts and Sciences - Arten_US
dc.contributor.departmentDepartment of Mathematicsen_US

Files in this item


This item appears in the following Collection(s)

  • Theses & Dissertations
    Items included under this collection are theses, dissertations, and project reports submitted as a requirement for completing a graduate degree at UPR-Mayagüez.

Show simple item record

CC0 1.0 Universal
Except where otherwise noted, this item's license is described as CC0 1.0 Universal