Smith Matrices For An U.F.D.

In this paper, we introduce the concepts of the great common P-divisor, least common P-multiple of two distinct nonzero elements x, y in arbitrary unique factorization domain (U.F.D.) R, and denote them by (x,y)p, and [x, y]p respectively. Let S = {x₁,……,x_n} be a set of n distinct nonzero non-associate elements in R, e≥1 an integer. The set S is called factor closed (FC), if for every x ∈S, every factor of x is still in S, or associates with an element x_j in S. The matrix ( S^e) = ( ( xi, xj)p) having the e-th power of the greatest common P-divisor ( xi, xj)p of xi and xj as its ( i ,j)-entry is called the e-th power GCD matrix on S. The matrix [S^e] = ([xi xj]pe) having the e-th power of the least common P-multiple [ xi ,xj ]p of xi and xj as its (i ,j)-entry is called the e-th power LCM matrix on 5. We obtained the following results:(1) ( S^e) is nonsingular for any set S;(2) If S is an FC set, then the determined of (S^e) has formula Det( Se) =Jpe(x1) ...Jpe), where the function Jpe is the generalized Jordan totient function; (3) A formula of the inverse of ( S^e) is given when S is an FC set; (4) If S is an FC set, then (Se) | [ S'].