Let be a ring, and let . A divisor of is an element such that there exists such that . The notation indicates that is a divisor of
Definition (greatest common divisor):
Let be a commutative ring, and let . A greatest common divisor is an element such that for all , and such that for any other element such that for all , we have .
Let be a commutative ring, and let . These elements are said to be coprime if and only if whenever is such that for all , then .
Proposition (a set of elements of a commutative ring divided by their greatest common divisor is coprime):