Molina-Salazar, Carlos A.

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    On the number of τ(n)-factor
    (2016) Molina-Salazar, Carlos A.; Ortiz-Albino, Reyes M.; College of Arts and Sciences - Sciences; Dziobiak, Stan; Ortiz Navarro, Juan; Department of Mathematics; Molina Bas, Omar
    In 2006, Anderson and Frazier defined the theory of τ-factorization as a generalization of the theory of comaximal factorizations given by McAdam and Swan. The theory of τ-factorization was built on integral domains. The idea can be associated with a restriction to the multiplicative operation. That is, they considered a symmetric relation τ on the nonzero nonunit elements of an integral domain and allowed two or more elements to be multiplied if and only if they were pairwise related. Formally, a nonzero nonunit element a of an integral domain D (denoted by D#) has a τ -factorization if a = λa1 ∗ · · · ∗ ak, where λ is an unit element and for any i#j, ai τ aj . In order to expand our vocabulary with respect to this concept, the ai ’s will be called τ -factors and a a τ -product of the ai ’s. This work studied a specific relation on the set of integers defined by Frazier and Anderson, and further studied by Hamon, Ortiz, Lanterman, Florescu, Serna and Barrios. Note that the notation and definition that will be used follows from Hamon’s work. They defined the relation τ(n) (also denoted as τn) on Z# by a τ(n) b if and only if a − b ∈ (n), the principal ideal generated by n. The main goal in this research was to find formulas that count the number of τ(n)-factors of an integer distinct to 0, 1 and -1. These formulas would be very useful tool to identify which elements cannot be τ(n)-factored properly.