## Publication: Ƭ-Mutiplicative sets

##### Authors
Ortiz-Albino, Reyes M.
##### College
College of Arts and Sciences - Sciences
##### Department
Department of Mathematics
M.S.
2014-05
##### Abstract
Motivated by McAdams and Swan work, Anderson and Frazier developed a theory called the theory of τ -f actorizations. It is a type of generalized factorization theory on integral domains. They used symmetric relations (denoted by τ ) on the set of nonunit nonzero elements of an integral domain D (denoted by D] ), in order to define what they called a τ -factorization of an element of D] . They called any factorization of an element x in D] of the form x = λx1 ·x2 · · · xn where xiτ xj for all i 6= j with 1 ≤ i, j ≤ n and λ ∈ U(D), a τ -f actorization of x. They classified the results based on three types of relations: divisive, multiplicative and associatedpreserving relations. This theory has been studied by Hamon (2007), Ortiz (2008), Reinkoester (2010) and Juett (2012). This work considered some preliminary definitions in order to study the theory of τ -factorizations with respect to an element, and further develops the most important types of relation of this theory. Several equivalences to the main types of definitions were obtained and then used to prove known results from other perspectives. In some cases, the results were more natural and their proofs were easier than the one provided in previous research works. It must be noted that this is the first attempt to try to understand what divisive, multiplicative and associatedpreserving relations mean. All the work previously done considered such type of relations, but the authors did not try to understand the nature of them. They used them only to prove theorems, because these relations provided a good behavior when the τ -factorizations were studied. Furthermore, this investigation studied some sets with specific properties with respect to a symmetric relation on D] and the connection with the τ -factorization theory. Finally, several examples were provided and some results were developed about the “τ -sets” on usual commutative ring properties.