Coprime Integers - Generalizations

Generalizations

Two ideals A and B in the commutative ring R are called coprime (or comaximal) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals (a) and (b) in the ring of integers Z are coprime if and only if a and b are coprime.

If the ideals A and B of R are coprime, then AB = A∩B; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

The concept of being relatively prime can also be extended to any finite set of integers S = {a₁, a₂, .... a_n} to mean that the greatest common divisor of the elements of the set is 1. If every pair in a (finite or infinite) set of integers is relatively prime, then the set is called pairwise relatively prime.

Every pairwise relatively prime finite set is relatively prime; however, the converse is not true: {6, 10, 15} is relatively prime, but not pairwise relative prime (each pair of integers in the set has a non-trivial common factor).