> 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.