TITLE: Groups and rings which are not sets
QUESTION [10 upvotes]: An algebraic structure such as a group, ring, field, etc. is usually defined to be a set with some operations satisfying certain properties.  I am curious what, if anything, goes wrong when the underlying collection of elements is not a set.  I don't have any particular example in mind, but I know some exist (in particular the surreals).  I'm just asking out of curiosity and I'm not sure where one would look up such a thing -- I am not a set theorist or logician by any stretch.
For a concrete question: what are the most elementary algebra results (on the level of Dummit and Foote, say) which can fail for groups or rings with "too many elements"?  Perhaps things like the existence of maximal ideals in commutative rings?

REPLY [5 votes]: One issue that might come up is that some standard constructions won't make sense, typically those that involve looking at equivalence classes.  For example, if you have a set group and you take a quotient by a subgroup, you're resulting object is a collection of equivalence classes, in this case a set of sets.  But when trying to do the same thing with a class group, you might end up with a class of classes, or a set of classes, which is obviously no good.  Scott's trick is usually provides a suitable work-around for this problem: when some of your equivalence classes are proper classes, use the elements of minimal (von Neumann) rank in the equivalence class in place of the entire equivalence class.