6.1/6.2, due on October 19

The most confusing part about the rest of section 6.1 was the notation note about the plus sign not meaning addition in congruence classes modulo I. I don't really think I understand what is is contained in those congruence classes. Is a+I a plus any possible i in I and that's why we write it in that form? Or is a+i a different congruence class than a+j for i,j in I, but we would write them both as a+I? Section 6.2 was a bit more confusing. It's a little strange that (a+I)+(c+I)=(a+c)+I. Why wouldn't this be plus 2I? Although I guess I wouldn't know what 2I really meant. How many congruence classes are there in R/I? Can we do an example of an Ideal and show all or most of it's congruence classes? The most confusing part of 6.2 has to be the First Isomorphism Theorem and it's proof. I thought I understood the discussion following the theorem but then the proof went over my head and the examples didn't really help me. It would be nice to discuss this part of the section.

Despite not really understanding how the First Isomorophism Theorem works, I think that is is really interesting that we can find always find a ring isomorphic to R as long as we have a surjective homomorphism between R and another ring. Suddenly it makes sense why we were concerned with surjective homomorphisms in an earlier chapter. I kept thinking that they weren't really very important because isomorphisms are so much more revealing. I am curious to know some of information that we can determine about R when given a homomorphism.