Vipul: New page: ==Statement== ===For a finite sequence=== Let $R$ be a commutative unital ring and $a_1, a_2, \dots, a_n \in R$ . An element $d \in R$ is termed a '''...

2009-01-24T01:45:01Z

New page: ==Statement== ===For a finite sequence=== Let <math>R</math> be a commutative unital ring and <math>a_1, a_2, \dots, a_n \in R</math>. An element <math>d \in R</math> is termed a '''...

New page

==Statement==

===For a finite sequence===

Let <math>R</math> be a [[commutative unital ring]] and <math>a_1, a_2, \dots, a_n \in R</math>. An element <math>d \in R</math> is termed a '''greatest common divisor''' or '''gcd''' of <math>a_1,a_2,\dots,a_n</math> if it satisfies the following equivalent conditions:

* <math>d|a_i</math> for all <math>1 \le i \le n</math> and if <math>c|a_i</math> for all <math>1 \le i \le n</math>, then <math>c | d</math>.
* <math>c|a_i</math> for all <math>1 \le i \le n</math> if and only if <math>c | d</math>.
* The ideal <math>(d)</math> is the intersection of all the [[principal ideal]]s of <math>R</math> containing <math>(a_1, a_2, \dots, a_n)</math>.

The greatest common divisor of a finite set of elements is not unique; if two elements are both greatest common divisors of <math>a_1,a_2, \dots, a_n</math>, then they are [[associate elements]].

===For any set===

Let <math>R</math> be a [[commutative unital ring]] and <math>S</math> be a subset of <math>R</math>. An element <math>d \in R</math> is termed a '''greatest common divisor''' of <math>S</math> if it satisfies the following equivalent conditions:

* <math>d|a</math> for all <math>a \in S</math>, and if <math>c|a</math> for all <math>a \in S</math>, then <math>c|d</math>.
* <math>c|a</math> for all <math>a \in S</math> if and only if <math>c|d</math>.
* The ideal <math>(d)</math> is the intersection of all the [[principal ideal]]s of <math>R</math> containing <math>S</math>.

==Facts==

* [[Greatest common divisors of the same subset form an associate class]]

Greatest common divisor - Revision history

Vipul: New page: ==Statement== ===For a finite sequence=== Let R be a commutative unital ring and a_1, a_2, \dots, a_n \in R. An element d \in R is termed a '''...

Vipul: New page: ==Statement== ===For a finite sequence=== Let $R$ be a commutative unital ring and $a_1, a_2, \dots, a_n \in R$ . An element $d \in R$ is termed a '''...