Characterization of Prime Ideals in (Z+,≤D)

A convolution is a mapping C of the set Z+ of positive integers into the set P (Z+) of all subsets of Z+ such that, for any n ∈ Z+ , each member of C(n) is a divisor of n. If D(n) is the set of all divisors of n, for any n, then D is called the Dirichlet’s convolution. Corresponding to any general convolution C, we can define a binary relation ≤C on Z+ by “m ≤C n if and only if m ∈ C(n)”. It is well known that Z+ has the structure of a distributive lattice with respect to the division order. The division ordering is precisely the partial ordering ≤D induced by the Dirichlet’s convolution D. In this paper, we present a characterization for the prime ideals in (Z+,≤D) , where D is the Dirichlet’s convolution.