Characterization of Prime Ideals in (Z+,<=D)

Sagi Sankar

Abstract

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

Keywords

Poset,Lattice,semi lattice,Convolution,ideal, prime ideal

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