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Publication:
Imagen de los τ-Productos

Imagen de los τ-Productos

##### 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

Relaciones,

Dominios de integridad,

Función,

Homomorfismo

##### Usage Rights

##### Persistent URL

Cite

Calderón Gómez, J. E. (2019).

*Imagen de los τ-Productos*[Thesis]. Retrieved from https://hdl.handle.net/20.500.11801/2482