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Journal of Mathematics and Statistics 5 (4): 342-347, 2009

ISSN 1549-3644

© 2009 Science Publications

Corresponding Author: A.N. El-Kassar, Lebanese American University, P.O. Box 13-5053, Chouran-Beirut, 1102 2801,

Lebanon Tel: 961-1-786456, Ext. 1189 Fax: 961-1-867098

342

Greatest Common Divisor and Least Common Multiple Matrices on

Factor Closed Sets in a Principal Ideal Domain

1

A.N. El-Kassar,

2

S.S. Habre and

3

Y.A. Awad

1

Lebanese American University, P.O. Box 13-5053,

Chouran-Beirut, 1102 2801, Lebanon

2

Lebanese American University, Lebanon

3

Lebanese International University, Lebanon

Abstract: Problem statement: Let T be a set of n distinct positive integers, x

1

, x

2

, ..., x

n

. 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. Many aspects

of arithmetics in the domain of natural integers can be carried out to Principal Ideal Domains (PID). In

this study, we extend many recent results concerning GCD and LCM matrices defined on Factor

Closed (FC) sets to an arbitrary PID such as the domain of Gaussian integers and the ring of

polynomials over a finite field. Approach: In order to extend the various results, we modified the

underlying computational procedures and number theoretic 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 the ring of polynomials over a finite field were given

to illustrate the new results. Conclusion: The extension of the GCD and LCM matrices to PIDs provided

a lager class for such matrices. Many of the open problems can be investigated in the new settings.

Key words: GCD matrix, lcm matrix, factor-closed sets, principal ideal domain

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

product

n

i

i 1

(x )

=

ϕ

∏

, where ϕ is Euler's totient phi-

function. Moreover, Smith considered the determinant

of the LCM matrix on a FC set and showed that it is

equal to the product

n

i i

i 1

(x ) (x )

=

ϕ π

∏

, 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

n

i

i 1

det[T] (x )

=

= ϕ

∏

. 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, gcd-

closed 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

J. Math. & Stat., 5 (4): 342-347, 2009

343

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,

1 2 i

1 2 i

b up p ...p

α α α

=

, 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

3

1 i

1 2 i

b up p ...p

α

α α

=

, then

1 2 i

1 2 i

q(b) (q(p )) (q(p )) ...(q(p ))

α α α

=

.

For a positive integer n, q(n) = n and for a Gaussian

integer a+bi, q(a+bi) = a²+b². Also, if f(x) is a

polynomial of degree n in Z

p

[x], then q(f(x)) = pⁿ.

Define φ

s

(b) = |U(S/<b>)|, the order of the group of

units U(S/<b>). Then φ

s

(b) ≥ 1 and the equality holds

iff b is a unit. Also, φ

s

(b) = q(p)

−1

iff p is prime in S. It

can be shown that φ

s

(b) is multiplicative and

if

3

1 i

1 2 i

b up p ...p

α

α α

=

, then:

1

2 i

1

S 1 1 2

1 1

2 i i

(b) (q(p ) 1)(q(p )) (q(p ) 1)

(q(p )) ...(q(p ) 1)(q(p ))

α −

α − α −

ϕ = − −

−

(1)

Now, if S = Z, then φ

s

(b) becomes Euler's phi-

function. Also, if

1 2 i

1 2 i

f(x) (p (x)) (p (x)) ...(p (x))

α α α

=

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:

ik k

k 1 k

i

r (

α

-1) r

sk 1

(f(x))=p (p 1)

=

=

∑

ϕ −

∏

(2)

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

1 2 i

k k k

1 2 i

u ...

β = β β β

, a product of distinct

Gaussian primes β

j

= a

j

+ib

j

, 1 ≤ j ≤ i, then:

( ) ( )

j

ik 1

2 2 2 2

s j j j j

j 1

( )= a + b a + b 1

−

=

φ β −

∏

For example, if β = 6+42i ~ 3(1+i)

3

(1+2i)

2

, then

φ

s

(β) = 640.

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

q(t

i

(x)) = q(t

i

(x)), where t

i

(x) ~xⁿ+a

n−1

xⁿ

−1

+...+a

1

x+a

0

,

t

j

(x) ~ xⁿ+b

n−1

xⁿ

−1

+...+b

1

x+b

0

, 0 ≤ a

j

, b

j

≤ p-1, j

0

is the

smallest index j such that a

j

≠ b

j

, then define t

i

(x) 〈t

j

(x)

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,

S S

d T d T'

(d) (d)

∈ ∈

φ = φ

∑ ∑

whenever T~ T′.

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

S

d E(b)

q(b) (d)

∈

= φ

∑

.

Proof: Let b ≠ 0. The result is true when b is a unit.

Suppose that b is a nonunit so that

1 2 i

1 2 i

b up p ...p

α α α

=

.

Since φ

s

is multiplicative, the function

S

d E( b)

f(b) (d)

∈

= ϕ

∑

is also multiplicative. For any prime element p

j

, (1)

gives:

n

j

n 0 1 n

j S S j S j S j

d E(p )

j j j j

n 1

j

n n

j j

f(p ) (d) (p ) (p ) ... (p )

1 (q(p ) 1) (q(p ) 1)(q(p )) ... (q(p ) 1)

(q(p ))

(q(p )) q(p )

∈

−

= ϕ = ϕ + ϕ + + ϕ

= + − + − + + −

= =

∑

J. Math. & Stat., 5 (4): 342-347, 2009

344

By the multiplicativity of f(b), we have

S

d E(b)

q(b) (d)

∈

= ϕ

∑

.

Corollary 1: (Euler's) If n is a positive integer, then

d 0,d|n

n (d)

>

= ϕ

∑

.

RESULTS

GCD matrices on FC sets in a PID: Throughout the

following, we consider T = {t

1

, t

2

, ..., t

n

} to be a q-

ordered 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 factor-

closed (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 factor-

closed or it is contained in a factor-closed set D.

Theorem 2: The GCD matrix [T] can be decomposed

into a product of an n×m matrix A and an m×n matrix

B, for some m ≥ n. The nonzero entries of A are equal

to φ

s

(d) for some d in a FC set D containing T and B is

an incidence matrix.

Proof: Let D = {d

1

, d

2

, ..., d

m

} be a FC set containing T

in the PID S Define the n×m matrix A = (a

ij

) by

s j j i

ij

(d ) if d E(t )

a

0 otherwise

ϕ ∈

=

and let B = (b

ij

) be the incidence

matrix corresponding to the transpose of A, where

ji

ij ji

1 if a 0

b

0 if a 0

≠

=

=

. Hence, the product AB is given by:

k i k i j

k j

k i j

n

ij ik kj s k s k

k 1 d E (t ) d E(t ) E( t )

d E(t )

s k i j

d E((t ,t ))

(AB) = a b = (d ) (d )

(d ) q((t ,t ))

= ∈ ∈ ∩

∈

∈

ϕ = ϕ =

ϕ =

∑ ∑ ∑

∑

Example 1: Let T = {1,1+x,1+x

3

,(1+x)

3

} in Z

2

[x].

Then [T] =

1 1 1 1

1 2 2 2

1 2 8 2

1 2 2 8

. Let D = {1, 1+x, (1+x)

2

,

1+x+x

2

, 1+x

3

, (1+x)

3

}. Then, [T]

4×4

= A

4×6

⋅B

6×4

where:

A =

1 0 0 0 0 0

1 1 0 0 0 0

1 1 0 3 3 0

1 1 2 0 0 4

and B =

1 1 1 1

0 1 1 1

0 0 0 1

0 0 1 0

0 0 1 0

0 0 0 1

Example 2: Let T = {1, 2, 5}. In Z[i],

1 1 1

[T] 1 4 1

1 1 25

=

.

Note that T is not FC in Z[i]. Select D = {1, 1+i, 2, 2+i,

1+2i, 5}. Then

1 0 0 0 0 0

A 1 1 2 0 0 0

1 0 0 4 4 16

=

and

111

0 1 0

0 1 0

B

0 0 1

0 0 1

0 0 1

=

.

Theorem 3: The GCD matrix [T] is the product of an

n×m matrix A and its transpose A

T

. The nonzero entries

of A are of the form

S

(d)

ϕ for some d in a FC set D

containing T.

Proof: Let D = {d

1

, d

2

, ..., d

m

} be a FC set containing

T. Define the n×m matrix A = (a

ij

) by

s j j i

ij

(d ) if d E(t )

a

0 otherwise

ϕ ∈

=

. Hence, the product A A

T

is

given by:

k i

k j

k i j k i j

n

Tij ik kj s k s k

k 1 d E (t )

d E(t )

s k s k i j

d E(t ) E( t ) d E((t ,t ))

(AA ) = a b = (d ) (d )

= (d ) = (d ) = q((t ,t ))

= ∈

∈

∈ ∩ ∈

ϕ ϕ

ϕ ϕ

∑ ∑

∑ ∑

Note that Theorem 2 and 3 hold even if T is not q-

ordered. In the case when both T and D are q-ordered,

B becomes in raw-echelon form.

Corollary 2: (Smith's Determinant over a PID) If T is

FC in S, then

[ ]

n

S i

i 1

det T (t ).

=

= ϕ

∏

Proof: Let T be a FC set. Choose D to be q-ordered and

D ~ T. From Theorem 2, the GCD matrix [T] = AB,

where A is an n×n lower triangular matrix and B is an

upper triangular matrix such that a

ii

= φ

s

(t

i

) and b

ii

= 1, 1

≤ i ≤ n. Therefore, det[T] = det[AB] = det[A]det[B] =

S 1 S 2 S n

(t ) (t )... (t )

ϕ ϕ ϕ

.

We note that if T′ =

1 n

i i2 i

{t ,t ,...,t }

is any

arrangement of the elements of T = {t

1

, t

2

, ..., t

n

} in S,

then det[T] = det[T′]. This can be verified as follows.

J. Math. & Stat., 5 (4): 342-347, 2009

345

The matrix [T] can be obtained from [T′] by switching

the rows and the columns of [T′]. Thus, [T] = E

1

E

2

...

E

i

[T′], where the E

j

′s are elementary matrices with

det[E

j

] = ±1, 1 ≤ j ≤ i. Hence, [T] and [T′] are similar

matrices and det[T] = det[T′].

Next, we consider the converse of Corollary 2. Let

S be a PID and let T = {t

1

, t

2

, ..., t

n

} be a nonempty set

of nonzero nonassociate elements in S with

[ ]

n

S i

i 1

det T (t ).

=

= ϕ

∏

Is it true that T is factor-closed in S?

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

(A)

ij

= ε

ij

S j

(t )

ϕ

, 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

1 2 n

(k ,k ,..., k )

M

denote the submatrix consisting of k

1th

, k

2th

,

... k

nth

columns of M.

Theorem 4: Let D = {t

1

, t

2

,...,t

n

, t

n+1

, ..., t

n+s

} be a

minimal FC set containing T = { t

1

, t

2

,...,t

n

} in S, where

t

1

〈 t

2

〈 ... 〈t

n

and t

n+1

〈 t

n+2

〈 ... 〈 t

n+s

. Then:

(

)

1 2 n 1 2 n

1 2 n

2

(k ,k ,...,k ) s k s k s k

1 k k ... k n s

det[T] det[E ] (t ) (t ).. (t )

≤ < < ≤ +

= ϕ ϕ ϕ

∑

Proof: Since [T] = AA

T

, Cauchy-Binet formula gives

that:

]

[

( )

( )

1 2 n 1 2 n

1 2 n

1 2 n

1 2 n

T

T

(k ,k ,...,k ) (k ,k ,...,k )

1 k k ... k n s

2

(k ,k ,...,k )

1 k k ... k n s

det T = det AA

det[A ]det[A ]

det[A ]

≤ < < ≤ +

≤ < < ≤ +

=

=

∑

∑

The result follows from the fact that:

1 2 n 1 2 n 1 2 n

(k ,k ,...,k ) (k ,k ,...,k ) s k s k s k

det[A det[E ] (t ) (t )... (t )

= ϕ ϕ ϕ

Corollary 3: Let [T] be the GCD matrix defined on T

in S. Then, det[T] ≥

s 1 s 2 s n

(t ) (t )... (t )

ϕ ϕ ϕ .

Proof: The terms in the summation of Theorem 4 are

nonnegative. Since the submatrix

1 2 n

(k ,k ,...,k )

E is lower

triangular with diagonal elements equal to 1, we have

that the term corresponding to (k

1

, k

2

, ..., k

n

) = (1, 2, ...,

n) is det[E(1, 2, ..., n)]

2

φ

s

(t

1

) φ

s

(t

2

)... φ

s

(t

n

) = φ

s

(t

1

)

φ

s

(t

2

)... φ

s

(t

n

). Therefore, det[T]≥

s 1 s 2 s n

(t ) (t )... (t )

ϕ ϕ ϕ .

Theorem 5: Let [T] be the GCD matrix defined on T in

S. Then, det[T] =

s 1 s 2 s n

(t ) (t )... (t )

ϕ ϕ ϕ if and only if T is

factor-closed in S.

Proof: The sufficient condition holds from Corollary 2.

Conversely, suppose that det[T] =

s 1 s 2 s n

(t ) (t )... (t )

ϕ ϕ ϕ .

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

(1,2,...r 1,n 1, r 1,...,n )

A

− + +

consisting of the 1

st

, 2

nd

, ..., (r

−1

)

th

, (n+1)

th

, (r+1)

th

, ...

and n

th

columns of A

n×(n+s)

is a lower triangular

matrix of nonzero determinant. Hence,

(1,2,...r 1,n 1, r 1,...,n )

E

− + +

is a {0, 1}-matrix whose diagonal

elements are equal to 1. Since

(1,2,...r 1,r 1,..., n, n 1)

E

− + +

can be

obtained from

(1,2,...r 1,n 1, r 1,...,n )

E

− + +

by performing a certain

numbers of successive column permutations,

(1,2,...r 1,r 1,...,n ,n 1)

det[E ]

− + +

= ±

(1,2,...r 1,n 1,r 1,...,n)

det[E ]

− + +

= ±1.

From Theorem 4, we have:

(

)

1 2 n 1 2 n

1 2 n

2

(k ,k ,...,k ) s k s k s k

1 k k ... k n s

s 1 s 2 s n s 1 s 2 s r 1 s r 1

s n s n 1 s 1 s 2 s n

det[T] det[E ] (t ) (t )... (t )

(t ) (t )... (t ) (t ) (t )... (t ) (t )

... (t ) (t ) ... (t ) (t )... (t )

≤ < < ≤ +

− +

+

= ϕ ϕ ϕ

= ϕ ϕ ϕ + ϕ ϕ ϕ ϕ

ϕ ϕ + > ϕ ϕ ϕ

∑

This contradicts the necessary condition that the

equality holds.

Inverses of GCD matrices in a PID: Let t be any

nonzero element in S. The generalized Mobius function

over S is defined by:

m

s

1 if tisa unit

(t) (-1) if tis the product of m nonassocia

te primes

0 otherwise

µ =

Note that:

s

d E( t)

1 if tisa unit

(d)

0 otherwise

∈

µ =

∑

(3)

Corollary 4: Let [T] be the GCD matrix defined on T

in S. Then, [T] is invertible and its inverse [T]

−1

= (r

ij

) is

given by:

J. Math. & Stat., 5 (4): 342-347, 2009

346

i k

j k

ij s k i s k j

t E (t ) s k

t E(t )

1

r (t / t ) (t / t )

(t )

∈

∈

= µ µ

ϕ

∑

Proof: Define the n×n matrices E = (e

ij

) and

U = (u

ij

) as follows:

i j

ij

1 if t E(t )

e

0 otherwise

∈

=

and

s i j i j

ij

(t / t ) if t E(t )

u

0 otherwise

µ ∈

=

. Then,

n

ij ik kj

k 1

(EU) e u

=

= =

∑

j k k i j

i j

s i j s k

t E (t ) t E (t /t )

1 if t ~t

(t / t ) (t )

0 otherwise

∈ ∈

µ = µ =

∑ ∑

. The last

equality follows from (3). Since the elements in T are

nonassociate, we have U = E

−1

. If D is the diagonal

matrix diag(φ

s

(t

1

), φ

s

(t

2

), ..., φ

s

(t

n

)) and A = ED

1/2

,

then [T] = AA

T

= (ED

1/2

)(ED

1/2

)

T

= EDE

T

. Therefore,

[T]

−1

= U

T

D

−1

U = (r

ij

), where r

ij

= (U

T

D

−1

U)

ij

=

i k

j k

n

ki kj s k i s k j

k 1 t E (t )

s k s k

t E(t )

1 1

u u (t / t ) (t / t )

(t ) (t )

= ∈

∈

= µ µ

ϕ ϕ

∑ ∑

.

Example 3: Let S = Z

2

[x] and let T = {1,1+x,1+x

2

,

(1+x)

3

,1+x+x

2

}, which is a q-ordered FC set of nonzero

nonassociate elements in Z

2

[x]. Then,

11111

1 2 2 2 1

[T]

1 2 4 4 1

1 2 4 8 1

1 1 1 1 4

=

. By corollary 4, [T]

−1

is obtained

as follows:

11 2

S S S

1 1 1 1 7

a 1 1

(1) (1 x) (1 x x ) 3 3

= + + = + + =

ϕ ϕ + ϕ + +

12 S

1

a 1

(1 x)

−

= = −

ϕ +

13 14 12 2

S

1 1

a a 0, a

(1 x x ) 3

− −

= = = =

ϕ + +

and so forth. Therefore:

1

7 / 3 1 0 0 1 / 3

1 3 / 2 1 / 2 0 0

[T]

0 1/ 2 3 / 4 1 / 4 0

0 0 1/ 4 1 / 4 0

1/ 3 0 0 0 1/ 3

−

− −

− −

=− −

−

−

.

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

)). It is clear that 1/[T] is

symmetric. Furthermore, permutations of the elements

of T yield similar reciprocal GCD matrices. For a

nonzero element t in T, define the function ξ

by

s

d E( t)

1

(t) q(d) (d)

q(t)

∈

ξ = µ

∑

. A generalized version of the

Mobius inversion formula can be used to show that

d E( t)

1

(d)

q(t)

∈

= ξ

∑

. Since ξ(t) is the product of two

multiplicative functions

1

q(t)

and

s

d E( t)

(t) q(d) (d)

∈

χ = µ

∑

,

we have that ξ(t) is itself multiplicative. Moreover, if p

is prime in S, then χ(p

n

) = 1q(p). Hence,

n

n

1 q(p)

(p )

(q(p))

−

ξ =

. Therefore:

S2

p E( t) p E(t )

1 (t)

(t) (1 q(p)) ( q(p))

q(t) (q(t))

∈ ∈

ϕ

ξ = − = −

∏ ∏

(4)

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.

Theorem 6: Let D = {d

1

, d

2

, ..., d

m

} be a FC set

containing T in S. The reciprocal GCD matrix defined

on T is the product of an n×m matrix A = (a

ij

), defined

by

j j i

ij

(d ) if d E(t )

a

0 otherwise

ξ ∈

=

and an m×n incidence matrix

B corresponding to A

T

.

Proof: Let A be as defined and let B be the m×n matrix

with

ij

ij ij

1 if a 0

b

0 if a 0

≠

=

=

. Then:

j i j i j

j j

j i j

m

ij ik kj j j

k 1 d E(t ) d E(t ) E( t )

d E(t )

j

d E ((t ,t )) i j

(AB) a b (d ) (d )

1

(d ) q((t ,t ))

= ∈ ∈ ∩

∈

∈

= = ξ = ξ

= ξ =

∑ ∑ ∑

∑

In a similar manner we prove the second

factorization given in the following theorem.

Theorem 7: Let D = {d

1

, d

2

, ..., d

m

} be a FC set

containing T in S and let C be the the n×n matrix given

by

j j i

ij

(d ) if d E(t )

a

0 otherwise

ξ ∈

=

. Then 1/[T] = CC

T

.

The proof of the following theorem is similar to

those of Theorems 4 and 5.

J. Math. & Stat., 5 (4): 342-347, 2009

347

Theorem 8: Let T be a set in S. Then, det(1/[T]) =

ξ(t

1

)ξ(t

2

)...ξ(t

n

) iff T is factor-closed in S.

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.

Theorem 9: If T is FC in S, then

[ ]

i

n

S i

i 1 p E(t )

det T (t ) ( q(p))

= ∈

= ϕ −

∏ ∏

.

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/[T]. Hence, [[T]] = D.(1/[T]).D, where D is the n×n

diagonal matrix with diagonal entries q(t

1

), q(t

2

), ...,

q(t

n

). From 4, we have that:

det[[S]] = det(D.(1/[T]).D) = det(D)²det(1/[T])

= (q(t

1

))²(q(t

2

))²...(q(t

n

))²ξ(t

1

)ξ(t

2

)...ξ(t

n

)

=

i

n

S i

i 1 p E(t )

(t ) ( q(p))

= ∈

ϕ −

∏ ∏

Cauchy Binet formula yields a formula for the

determinant of the LCM matrix defined on a set T

which is not necessarily FC. The formula is given by:

( )

( )

1 2 n

1 2 n

1 2 n 1 2 n

2

(k ,k ,...,k )

1 k k ... k n s

2

k k k k k k

det[[T]] det E

q(t )q(t )...q(t ) (t ) (t )... (t )

≤ < < ≤ +

=

ξ ξ ξ

∑

From Theorem 9, we have that the determinant of

the GCD matrix on S divides the determinant of the

LCM matrix whenever T is FC in S.

DISCUSSION

Most of the existing results related to GCD and

LCM matrices are obtained in the domain of natural

integers. The results are based on certain number

theoretic functions such as Euler’s phi function and the

Mobius function. These function and their properties

can be generalized to principal ideal domains. By

describing the underlying computational procedures

and the various properties in the new settings, the

existing results related to GCD and LCM defined on

factor closed sets are extended to PIDs. This provides a

large class of such matrices where many new examples

can be constructed. In particular, examples in the

domains of Gaussian integers and the ring of

polynomials over a finite field may give new insight to

some open problems.

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.

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