6.1

1. Ideals. It doesn't seem that bad. It's always harder to broaden my definition of things, but that's the nature of generalization. I mean, that's what we have to do, right? We can't conceive of everything and we can only conceive of things pretty close to our experience. So we generalize. We make sweeping claims from finitely many specifics. And proofs are generalizations too. But, uh, nothing difficult here. So I either really get it or I really don't.

2. I think it's interesting that even though we are broadening our definitions, congruence is still based on subtraction. I know that we could define that symbol to mean a lot of things (redefining addition and satisfying our axioms, of course), but it seems weird to me that we don't have a lot of different flavors of congruence (maybe we do...?) like difference congruence, logarithmic congruence, or cross-product congruence or something.