Greatest Common Divisor and Least Common Multiple Matrices on Factor Closed Sets in a Principal Ideal Domain

Problem statement: Let T be a set of n distinct positive integers, x 1, x2, ..., xn. The n×n matrix [T] having (xi, xj), the greatest common divisor of x i and xj, as its (i,j)-entry is called the greatest common divisor (GCD) matrix on T. The matr ix [[T]] whose (i,j)-entry is [xi, xj], the least common multiple of x i and xj, is called the least common multiple (LCM) matrix on T. Many aspects of arithmetics in the domain of natural integers can be carried out to Principal Ideal Domains (PID) . n this study, we extend many recent results concernin g GCD and LCM matrices defined on Factor Closed (FC) sets to an arbitrary PID such as the do main of Gaussian integers and the ring of polynomials over a finite field. Approach: In order to extend the various results, we modifie the underlying computational procedures and number theo retic functions to the arbitrary PIDs. Properties of the modified functions and procedures were given in the new settings. Results: Modifications were used to extend the major results concerning GCD and LCM matrices defined on FC sets in PIDs. Examples in the domains of Gaussian integers and th e ring of polynomials over a finite field were give n to illustrate the new results. Conclusion: The extension of the GCD and LCM matrices to PIDs p rovided a lager class for such matrices. Many of the open p roblems can be investigated in the new settings.


INTRODUCTION
Let T = {x 1 , x 2 , ..., x n } be a set of n distinct positive integers. The n×n matrix [T] having (x i , x j ), the greatest common divisor of x i and x j , as its (i,j)-entry is called the Greatest Common Divisor (GCD) matrix on T. The matrix [[T]] whose (i,j)-entry is [x i ,x j ], the least common multiple of x i and x j , is called the least common multiple (LCM) matrix on T. The set T is said to be factor closed (FC) if it contains every divisor of x for any x∈T. In 1876, Smith [11] showed that the determinant of the GCD matrix [T] on a FC set T is the , where π is a multiplicative function defined for a prime power p r by π(p r ) = −p. Since then many papers related to Smith's results have been published. Recently, this field has been studied intensively. This new inspiration started in by Beslin and Ligh [3,4] . In [3] , Beslin and Ligh obtained a structure theorem for GCD matrices and showed that, if S is FC, then . They conjectured that the converse is true. In [10] , Li proved the converse and provided a formula for the determinant of an arbitrary GCD matrix. Beslin and Ligh [4,5] generalized these results by extending the FC sets to a larger class of sets, gcdclosed sets. In [1] , a structure theorem for [ [T]] was obtained from the structure of the reciprocal GCD matrix 1/[T], the (i,j)-entry of which is 1/(x i , x j ). Given a FC set T, Bourque and Ligh [6]

calculated the inverses of [T] and [[T]] and showed that [[T]][T]
−1 is an integral matrix. In that study, they stated their famous conjecture that the LCM matrix on any gcd-closed set is invertible. Bourque and Ligh [7,8] investigated the structures, the determinants and the inverses associated with classes of arithmetical functions. For a brief review of papers relating to Smith's determinant, we refer to [9] . Using the language of posets, the authors gave a common structure that is present in many extensions of Smith's determinants. Beslin and El-Kassar [2] extended the results in [3] to unique factorization domains. The purpose of this study is to extend many of the recent results concerning GCD and LCM matrices defined on factor-closed sets to arbitrary Principal Ideal Domains (PID) such as the domain of Gaussian integers and the ring of polynomials over a finite field.

MATERIALS AND METHODS
Let S be a PID and let a, b∈S. We say that a and b are associates and write a~b, if a = ub for some unit element u in S. If b is a nonzero nonunit element, then b has a unique factorization, up to associates, into prime elements in S. That is, , where the p j ′s are distinct primes in S. Also, every finite set {b 1 , b 2 , ..., b n } admits, up to associates, a greatest common divisor. For a nonzero element b in S, define q(b) to be |S/<b>|, the order of the quotient ring S/<b>, where <b> is the principal ideal generated by b. Note that q(u) = 1, for any unit u. Also note that in Z, Z[i] and Z p [x], q(b) is finite ∀ b ≠ 0. Throughout the following we consider S to be a PID having the property that q(b) is finite ∀ b ≠ 0. It can be shown that q(ab) = q(a)q(b). Hence, if Now, if S = Z, then φ s (b) becomes Euler's phifunction. Also, if 1 2 i is a polynomial of degree n in S = Z p [x], a product of powers of distinct irreducible polynomials p j (x), 1 ≤ j ≤ i, then: where, p j (x) is of degree r j . For example, if f(x) = (x²+ 1) 4 (x+1) 4 (x 3 +x 2 +x+4) in S = Z 2 [x], then φ s (f(x)) = 3 7 . 2 5 .13. Now, if Gaussian primes β j = a j +ib j , 1 ≤ j ≤ i, then: Let S be a PID and let T = {t 1 , t 2 , ..., t n } be a set of nonzero nonassociate elements in S. Define a linear ordering 〈 on T according to the following scheme: If q(t i ) < q(t j ) then t i 〈 t j and if the equality q(t i ) = q(t j ) holds then order t i and t j according to any scheme depending on the given domain S. For instance, if S = Z[i] and q(t i ) = q(t j ), where t i ~ a+ib, t j ~ c+id, a, b, c, d ≥ 0, then define t i 〈 t j whenever b < d. In the case S = Z p [i] and whenever a j0 <b j0 . If the set T is ordered so that t 1 〈t 2 〈...〈t n , we say that T is q-ordered. Two sets T and T′ in S are associates, denoted by T~ T′, iff each element in T is associate to an element in T′ and vice versa. For a nonzero element b, let E(b) be a complete set of distinct nonassociate divisors of b in S. Then, E(a)∩E(b) ~E((a, b)) and E(p m ) ~{1, p 1 , p 2 ,..., p m }. Note that if t~t′, then q(t) = q(t′) and φ s (t) = φ s (t′). Also, Theorem 1: Let S be a PID and let b be a nonzero element in S. If E(b) is a complete set of distinct nonassociate divisors of b, then Suppose that b is a nonunit so that Since φ s is multiplicative, the function ∑ is also multiplicative. For any prime element p j , (1) gives:

GCD matrices on FC sets in a PID:
Throughout the following, we consider T = {t 1 , t 2 , ..., t n } to be a qordered set of nonzero nonassociate elements of a PID S Define the GCD matrix on S to be the n×n matrix [T] = (t ij ) = q((t i , t j )). The set T is said to be a factorclosed (FC) in S iff t i ∈T and d|t i implies that d~ t j for some t j ∈T. Note that any set T in S is either factorclosed or it is contained in a factor-closed set D.
. Hence, the product AB is given by: .
1+2i, 5}. Then . Hence, the product A A T is given by: Note that Theorem 2 and 3 hold even if T is not qordered. In the case when both T and D are q-ordered, B becomes in raw-echelon form.  Consider a minimal FC set D = {t 1 , t 2 , ..., t n , t n+1 , ..., t n+r } containing T = { t 1 , t 2 , ..., t n } with t 1 〈 t 2 〈 ... 〈 t n and t n+1 〈 t n+2 〈 ...〈 t n+r . Define an n×(n+r) matrix A by , where ε ij is 1 if t j ∈E(t i ) and 0 otherwise. Denote the matrix (ε ij ) n×(n+r) by E, a {0,1}matrix. Note that the matrix A is the same matrix A defined in Theorem 3.
For an n×m matrix M, n > m and any set of indices k 1 , k 2 , ..., k n with 1 ≤ k 1 < k 2 < ... < k n ≤ m, let . For contradiction purposes, suppose that T is not FC. Let D = {t 1 , t 2 , ..., t n , t n+1 , ..., t n+s } be a minimal FC set containing T in S such that t 1 〈 t 2 〈 ... 〈 t n and t n+1 〈t n+2 〈 ... 〈 t n+s . Since T is not FC, D is not associate to T in S. Then, t n+1 is in D but not in T and t n+1 ∈E(t) for some t in T. Now, let t r be the first element in T such that t n+1 ∈E(t r ). Then, the submatrix if t is a unit (t) (-1) if t is the product of m nonassociate primes 0 otherwise

Reciprocal GCD Matrices in a PID:
The reciprocal GCD matrix on T in S is the n×n matrix 1/[T] whose (i,j)-entry is 1/q((t i , t j )). . Therefore: where, the product runs over all prime divisors p of t in E(t).
In the following two theorems we obtain two factorizations for the reciprocal GCD matrices. . Then: In a similar manner we prove the second factorization given in the following theorem. The proof of the following theorem is similar to those of Theorems 4 and 5.

LCM Matrices on FC Sets in a PID:
The least common multiple (LCM) matrix defined on T in S is the n×n matrix [[T]] = (t ij ), where t ij = q([t i , t j ]) and [t i , t j ] is the least common multiple of t i and t j in S.
Proof: Since [t i , t j ]~(t i t j )/( t, i t j ), we have and q([t i , t j ]) = q(t i )q(t j )/q(( t, i t j )). Now q(t i ) can be factored out from the i th row and q(t j ) from the j th column to obtain 1/

CONCLUSION
The extension of the GCD and LCM matrices to PIDs provide a lager class for such matrices. Many of the open problems can be investigated in the new settings. For future study, we suggest the problem of extending GCD and LCM matrices defined on gcd closed sets to PIDS.