3 ms·

> You can extend the concepts of gcd, primality, divisibility, etc to rings in general Thats true, but you have a "real" GCD only in special types of rings (pr

by vog 13y ago

> You can extend the concepts of gcd, primality, divisibility, etc to rings in general Thats true, but you have a "real" GCD only in special types of rings (principal ideal domains). In other rings, you only have Ideals as rough generalization of GCDs. > Therefore, if a and b are integers, (a, b) is the set of multiples of their gcd - just like (3) is the set of multiples of three. To make this more clear, in the notation of Ideals, you can write this: (15,6) = (3) That is, the Ideal generated by 15 and 6 is same as the Ideal generated by 3. And for nonnegative integers, this essentially means the same as saying that 3 is the GCD of 15 and 6: gcd(15,6) = 3 There is still some "unclean" step involved here (that is, identifying numbers by their principal ideal, i.e. treating 3 and (3) as if these were equal), but I think this justifies the notation nevertheless.

PieSquared 13y ago
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

by vog 13y ago

PieSquared 13y ago

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