You're is exactly right. To add to what you were saying and expand a little bit: A principal ideal domain is a ring in which every ideal is generated by only o

You're is exactly right. To add to what you were saying and expand a little bit: A principal ideal domain is a ring in which every ideal is generated by only one element, so whenever we see (a, b), we know there is some element c such that (a, b) = (c). I think this is what vog meant by having a "real" GCD - only in a principal ideal domain is your gcd unique. Without uniqueness, we can still define a greatest common divisor such that if gcd(a, b) = g, we know that there is nothing we can multiply by g to get a divisor of both a and b; that is, there's no extra factor we can add to g in order to get another factor of both a and b. That is enough to call g a GCD - but it's not necessarily unique! It turns out that in rings which aren't principal ideal domains, you can have more than one GCD! It's bizarre to think exactly what "greatest" means in this context, but you can also just think of it as "can't add any more factor while still dividing both a and b". Ring theory is fun! (And practical, sometimes - you can describe some algorithms very elegantly via embedding the things you're working with in unusual rings.) More info, with some examples and counterexamples: https://en.wikipedia.org/wiki/Principal_ideal_domain