GCD Matrices Defined on GCD-Closed Sets in Principal Ideal Domains

Abstract

Let S = {x1, x2, ..., xn} be a set of n distinct positive integers. The matrix [S] = (sij) having the greatest common divisor (xi, xj) of xi and xj as its i, j-entry is called the greatest common divisor (GCD) matrix on S. Beslin and Ligh obtained a structure theorem for GCD matrices and generalized Smith’s determinant to factor-closed sets and gcd-closed sets. In a previous paper, we extended many of the results concerning the GCD matrices defined on factorclosed sets to principal ideal domains such as the domain of Gaussian integers and the rings of polynomials over the finite field. In this paper, we extend these results to GCD matrices defined on gcd-closed sets in a principal ideal domain.