The set of k-units modulo n

Let R be a ring with identity, U(R) be the group of units of R and k be a positive integer. We say that a is an element of U(R) is ak-unit if alpha(k) = 1. In particular, if the ring R is Z(n) for some positive integer n we say that alpha is a k-unit modulo n. We denote by U-k(n) the set of k-units modulo n. We represent the number of k-units modulo n by du(k)(n) and the ratio of k-units modulo n by rdu(k)(n)= phi(n)/du(k)(n), where phi is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of the equation rdu(2)(n) = 1 are the divisors of 24. Our main result finds all positive integers n such that rdu(k)(n) = 1 for a given k. Then we connect this equation with the Carmichael numbers and two of their generalizations, namely, Knodel numbers and generalized Carmichael numbers.