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惟一因子分解整环上的GCD幂矩阵与LCM幂矩阵

Note on Power GCD and LCM Matrices for U.F.D.
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摘要 设S={x1,x2,…,xn}是惟一分解整环R上的不同元素构成的集合,e≥1是一个正整数.(xi,xj)和[xi,xj]分别表示xi,xj的最大公因子和最小公倍数.S称为因子封闭集(简称FC集),如果对S中的任何元xi,它的任意一个因子是S中的一个元的相伴元.以(xi,xj)的e次方为i行j列元素的矩阵称为GCD幂矩阵,记为(Se);以[xi,xj]的e次方为i行j列元素的矩阵称为LCM幂矩阵,记为[Se].作者证明了若S是FC集,则(Se)整除[Se],即[Se]等于(Se)与R上另一个矩阵的乘积,推广了Bourque和Ligh在1992年所得的结果. Let S={x1,…,xn} be a set of n distinct elements in a unique factorization domain R and e≥1 an integer, the set S is called factor closed (FC), if x∈S, x is an associate of one element in S. The matrix (Se)=((xi,xj)e) having the e-th power (xi,xj)e of the greatest common divisor (GCD) of xi and xj as its (i,j) entry is called the e-th power GCD matrix on S; The matrix [Se]=([xi,xj]e) having the e-th power [xi,xj]e of the least common multiple (LCM) of xi and xj as its (i,j) entry is called the e-th power LCM matrix on S. In this paper, the authors prove that if S is FC, then the e-th power GCD matrix (Se) on S divides the e-th power LCM matrix [Se] on S in ring Mn(R) of n×n matrices over the U.F.D. R. This extends the result of Bourque and Ligh in 1992 which considers the usual positive integer case.
出处 《四川大学学报(自然科学版)》 CAS CSCD 北大核心 2005年第2期417-419,共3页 Journal of Sichuan University(Natural Science Edition)
关键词 GCD幂矩阵 LCM幂矩阵 因子 UFD power GCD matrix power LCM matrix factorization UFD
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参考文献11

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