Greatest Common Divisor and Least Common Multiple Matrices on Factor Closed Sets in a Principal Ideal Domain

Abstract

Problem statement: Let T be a set of n distinct positive integers, x₁, x₂, ..., x_n. The n×n matrix [T] having (x_i, x_j), the greatest common divisor of x_i and x_j, as its (i,j)-entry is called the greatest common divisor (GCD) matrix on T. The matrix [[T]] whose (i,j)-entry is [x_i, x_j], the least common multiple of x_i and x_j, is called the least common multiple (LCM) matrix on T. Many aspects of arithmetics in the domain of natural integers can be carried out to Principal Ideal Domains (PID). In this study, we extend many recent results concerning GCD and LCM matrices defined on Factor Closed (FC) sets to an arbitrary PID such as the domain of Gaussian integers and the ring of polynomials over a finite field. Approach: In order to extend the various results, we modified the underlying computational procedures and number theoretic functions to the arbitrary PIDs. Properties of the modified functions and procedures were given in the new settings. Results: Modifications were used to extend the major results concerning GCD and LCM matrices defined on FC sets in PIDs. Examples in the domains of Gaussian integers and the ring of polynomials over a finite field were given to illustrate the new results. Conclusion: The extension of the GCD and LCM matrices to PIDs provided a lager class for such matrices. Many of the open problems can be investigated in the new settings.