Recall that one may define relative primality of a pair of elements in full generality: let be a monoid, and let and be two elements of . We say and are relatively prime if all common divisors of and are units. Formally: .
However, we will work in the more restricted setting of a commutative ring with identity. As we get more and more specialized types of rings, we get better and better characterizations of relative primality.
In integral domains. In this setting, the divisibility relation is more meaningful, and we have: two elements and are relatively prime if, and only if, for any element , the containement implies . In effect, this says the ideal generated by and is “bigger” than any non-trivial principal ideal in the ring.
In GCD domains. Recall that a GCD domain is an integral domain where a greatest common divisor for any pair of elements is guaranteed to exist. In other words, there is always a unique minimal principal ideal containing . Together with the previous characterization in integral domains, this yields: two elements and are relatively prime if, and only if, a greatest common divisor of and is .
In unique factorization domains. Two elements and are relatively prime if, and only if, for every irreducible element , we fail to have the containement . For this to work, we in fact only need the existence of at least one irreducible factor for every non-zero non-unit element, so this characterization could work in more general types of rings.
In principal ideal domains. Now the situation is maybe as good as it gets, theoretically speaking. Two elements and are relatively prime if, and only if, the two ideals and are comaximal, i.e. . This is a strenghtening of the previous characterization on GCD domains (recall that every PID is a UFD, hence is a GCD domain). Of course, is just a fancier way of writing . Most of the time, the equation is called Bézout’s Identity.