Publication:
Imagen de los τ-Productos

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Authors
Calderón Gómez, José E.
Embargoed Until
Advisor
Ortiz Albino, Reyes M.
College
College of Arts and Sciences - Art
Department
Department of Mathematics
Degree Level
M.S.
Publisher
Date
2019-05-15
Abstract
The 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.
Keywords
factorizaciones,
Relaciones,
Dominios de integridad,
Función,
Homomorfismo
Usage Rights
Except where otherwise noted, this item’s license is described as Attribution-NonCommercial-NoDerivatives 4.0 International
Cite
Calderón Gómez, J. E. (2019). Imagen de los τ-Productos [Thesis]. Retrieved from https://hdl.handle.net/20.500.11801/2482