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 a k = 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 U k (n) the set of k-units modulo n.By du k (n) we represent the number of k-units modulo n and with rdu k (n) = φ(n) du k (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 rdu k (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.S. K. Chebolu [2] proved that in the ring Z n the square of any unit is 1 if and only if n is a divisor of 24.This property is known as the diagonal property for the ring Z n .Later, K. Genzlinger and K. Lockridge [7] introduced the function du(R), which is the number of involutions in R (that is the elements in R such that a 2 = 1), and provided another proof to Chebolu's result about the diagonal property.The diagonal property also has been studied in other rings.For instance, S. K. Chebolu [4] found that the polynomial ring Z n [x 1 , x 2 , . . ., x m ] satisfies the diagonal property if and only if n is a divisor of 12 and S. K. Chebolu et al. [3] also characterized the group algebras that verifies this property.Let R be a ring with identity and U (R) the group of units of R. The aim of this paper is to study the elements of a ring with the following property: for a given k ∈ Z + , we say that an element a in U (R) is a k-unit if a k = 1.So, we ask for the number of this elements, and for that we extend the definitions of the functions given by K. Genzlinger and K. Lockridge [7], particularly du k (R) will represent the number of k-units of R.Here we present a formula for this function when U (R) can be expressed as a finite direct product of finite cyclic groups and when R = Z n .Furthermore, we study the case when R = Z n and each unit is a k-unit.Previously, as mentioned before, this problem has been considered when k = 2 and more generally for fields and group algebras when k is a prime number, see [3].In the other hand, a well studied topic in number theory are the Carmichael's numbers, which in terms of the k-unit concept, are composite positive integers such that any unit is an (n -1)-unit.Here, we find some connections between the equation rdu k (n) and the concepts of Knödel and generalized Carmichael numbers.In the sequel, for x an element of a group G, by |x| we denote the order of x.Besides, for a prime number p and a positive integer n, the symbol ν p (n) means the exponent of the greatest power of p that divides n, gcd(a, b) denotes the greatest common divisor of a and b, and φ is the Euler's totient function.If A = {a 1 , a 2 , . . ., a n } and f is a defined function on A, we writeand when A = ∅, we assume that a∈A f (a) = 1. Set of k-units of a ringIn this section we give some definitions and get some preliminary results.