I'm just not really sure what well-definition implies. I think I get it, but I could use a little feedback.
Yes, that is correct. Well-defined means if you pick two things from the same class and multiply (or do any other operation) you will always get the same answer - it doesn't matter which element from the class you pick. So, whenever you see something whose elements are of the form or or where or are some sort of algebraic structure (for instance, an ideal or a ring, a normal subgroup of a group) you would need to take two elements and see if they are always mapped to the same thing.
For instance, let and let be the set of equivalence classes of the integers modulo n, and again let the operation be addition.
Can you show that is well-defined?