gcd domain

gcd domain

Throughout this entry, let

D

be a

commutative ring

with

1

≠

0

.

A gcd (

greatest common divisor

) of two

elements

a

,

b

∈

D

, is an element

d

∈

D

such that:

1.

d

∣

a

and

d

∣

b

,

2.

if

c

∈

D

with

c

∣

a

and

c

∣

b

, then

c

∣

d

.

For example,

0

is a gcd of

0

and

0

in any

D

. In fact, if

d

is a gcd of

0

and

0

, then

d

∣

0

. But

0

∣

0

, so that

0

∣

d

, which means that, for some

x

∈

D

,

d

=

0

⁢

x

=

0

. As a result,

0

is the unique gcd of

0

and

0

.

In general, however, a gcd of two elements is not unique. For example, in the

ring of integers

,

1

and

-

1

are both gcd’s of two relatively

prime elements

. By definition, any two gcd’s of a pair of elements in

D

are

associates

of each other. Since the

binary relation

“being associates” of one anther is an

equivalence relation

(

not

a

congruence relation

!), we may define

the

gcd of

a

and

b

as the set

GCD

⁡

(

a

,

b

)

:=

{

c

∈

D

∣

c

⁢

is a gcd of

⁢

a

⁢

and

⁢

b

}

,

For example, as we have seen,

GCD

⁡

(

0

,

0

)

=

{

0

}

. Also, for any

a

∈

D

,

GCD

⁡

(

a

,

1

)

=

U

⁡

(

D

)

, the

group of units

of

D

.

If there is no confusion, let us denote

gcd

⁡

(

a

,

b

)

to be any element of

GCD

⁡

(

a

,

b

)

.

If

GCD

⁡

(

a

,

b

)

contains a unit, then

a

and

b

are said to be

relatively prime

. If

a

is

irreducible

, then for any

b

∈

D

,

a

,

b

are either relatively prime, or

a

∣

b

.

An

integral domain

D

is called a

gcd domain

if any two elements of

D

, not both zero, have a gcd. In other words,

D

is a gcd domain if for any

a

,

b

∈

D

,

GCD

⁡

(

a

,

b

)

≠

∅

.

Remarks

•

A

unique factorization domain

, or UFD is a gcd domain, but the

converse

is not true.

•

A

Bezout domain

is always a gcd domain. A gcd domain

D

is a Bezout domain if

gcd

⁡

(

a

,

b

)

=

r

⁢

a

+

s

⁢

b

for any

a

,

b

∈

D

and some

r

,

s

∈

D

.

•

In a gcd domain, an

irreducible element

is a prime element.

•

A gcd domain is

integrally closed

. In fact, it is a

Schreier domain

.

•

Given an integral domain, one can similarly define an lcm of two elements

a

,

b

: it is an element

c

such that

a

∣

c

and

b

∣

c

, and if

d

is an element such that

a

∣

d

and

b

∣

d

, then

c

∣

d

. Then, a

lcm domain

is an integral domain such that every pair of elements has a lcm. As it turns out, the two notions are

equivalent

: an

integral domain is lcm iff it is gcd

.

The following diagram indicates how the different domains are related:

Euclidean domain

(

http://planetmath.org/EuclideanRing

)

⟹

PID

⟹

UFD

⇓

⇓

Bezout domain

⟹

gcd domain

References

1

D. D. Anderson,

Advances in Commutative Ring Theory:

Extensions

of

Unique Factorization

, A Survey

, 3rd Edition, CRC Press (1999)

2

D. D. Anderson,

Non-Noetherian Commutative Ring Theory: GCD Domains, Gauss’ Lemma, and

Contents of Polynomials

, Springer (2009)

3

D. D. Anderson (editor),

Factorizations

in Integral Domains

, CRC Press (1997)

Title

gcd domain

Canonical name

GcdDomain

Date of creation

2013-03-22 14:19:51

Last modified on

2013-03-22 14:19:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

26

Author

CWoo (3771)

Entry type

Definition

Classification

msc 13G05

Related topic

GreatestCommonDivisor

Related topic

BezoutDomain

Related topic

DivisibilityInRings

Defines

gcd

Defines

greatest common divisor

Defines

relatively prime

Defines

lcm domain

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