The set of k-units modulo n

Let R be a ring with identity, U(R) the group of units of R and k a positive integer. We say that a ∈ U(R) is k-unit if ak = 1. Particularly, if the ring R is Zn, for a positive integer n, we will say that a is a k-unit modulo n. We denote with Uk(n) the set of k-units modulo n. By duk(n) we represent the number of k-units modulo n and with rduk(n) = φ(n) duk(n) the ratio of k-units modulo n, where φ is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation rdu2(n) = 1 are the divisors of 24. The main result of this work, is that for a given k, we find the positive integers n such that rduk(n) = 1. Finally, we give some connections of this equation with Carmichael’s numbers and two of its generalizations: Knödel numbers and generalized Carmichael numbers.

• Autores: Caranguay Mainguez, Jhony Fernando; Castillo Gómez, John Hermes
• Tipo de recurso: Article of investigation
• Fecha de publicación: 2022
• Institución: Universidad de Antioquia
• Repositorio: Repositorio UdeA
• Idioma: eng
• OAI Identifier: oai:bibliotecadigital.udea.edu.co:10495/46174
• Acceso en línea: https://hdl.handle.net/10495/46174
• Palabra clave: Teoría de los números; Numbers, Theory of; Álgebra; Algebra; Conjunto unitario de un anillo; Conjunto unitario de un anillo; Unidad K; Número de carmichael; Número de Knödel; Diagonal property; K-unit; Carmichael number; Knödel number; Propiedad diagonal
• Rights: openAccess
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/