Isn’t it true that commutative algebra is contained within set theory? Aren’t all the objects sets?
Sets are one way (among others) to encode such mathematical structures. This doesn’t mean these structures are sets. For example, is the empty set an element of pi? This question is meaningless because the answer would have to depend on the encoding being used, and not on the properties of pi itself. It’s imposing additional structure above and beyond that of the mathematical structure in question.
See here for more discussion: https://news.ycombinator.com/item?id=16080027.
No. It is a set together with additional structure. Without this additional structure you just have a set, not a ring.
> So is a group.
Again, no. It is a set together with additional structure.
Note: The collection of all groups is not even a set!