使得幂GCD阵(S^e)整除幂LCM矩阵[S^e]的四元gcd封闭集S的一个刻画(英文)

A characterization of the gcd-closed set S with |S|=4 such that(S^e) divides |S^e|

Hong在2002年证明了如下结果:若S为gcd封闭集且|S|3,则在|S|阶整数矩阵环M|S|(Z)中,GCD矩阵(S)整除LCM矩阵[S].设e 1为给定的整数.在本文中,我们给出了关于四元gcd封闭集S的充分必要条件,使得在环M4(Z)中,定义在S上的e次幂GCD矩阵(Se)整除e次幂LCM矩阵[Se].这部分解决了Hong在2002年提出的一个公开问题.

In 2002, Hong showed that for any gcd-closed set S with ｜ S ｜≤3, the GCD matrix （S） on S divides the LCM matrix [S] on S in the ring M｜s｜ （Z） of ｜ s｜ ×｜ s｜ matrices over the integers. Let e≥ be an arbitrary given integer. In this paper, a necessary and sufficient conditions on the ged closed set S with ｜ S ｜ = 4 such that the power GCD matrix （S^e）on S divides the power LCM matrix [ S^e ] on S in the ring M4 （Z） of 4 × 4 matrices over the integers is proved. This solves partially an open question raised by Hong in 2002.