# Commutative Ring Theory/Greatest common divisors

**Definition (divisor)**:

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 .

**Definition (coprime)**:

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)**:

Let