Non-divisibility of LCM matrices by GCD matrices on gcd-closed sets

摘要

In this paper, we consider the divisibility problem of LCM matrices by GCD matrices in the ring M-n(Z) proposed by Shaofang Hong in 2002 and in particular a conjecture concerning the divisibility problem raised by Jianrong Zhao in 2014. We present some certain gcd-closed sets on which the LCM matrix is not divisible by the GCD matrix in the ring M-n(Z). This could be the first theoretical evidence that Zhao's conjecture might be true. Furthermore, we give the necessary and sufficient conditions on the gcd-closed set S with vertical bar S vertical bar <= 8 such that the GCD matrix divides the LCM matrix in the ring. M-n(Z) and hence we partially solve Hong's problem. Finally, we conclude with a new conjecture that can be thought as a generalization of Zhao's conjecture. (C) 2016 Elsevier Inc. All rights reserved.