I was very confused by the several of the proofs in this section. And actually by reasoning through a bunch of the terms in this section too. The whole concept of a maximal ideal is confusing to me and I couldn't follow the proof. Another thing I was confused with was an example of a prime ideal on page 6. They used the zero ideal as an example but I thought part of the definition was that p couldn't be 0 (earlier on the page it says "if p not 0 or plus minus one, then p is prime if and only if whenever...). I feel like this is all one level too abstract for me to wrap my head around (not that I don't find it interesting when I do understand what is going on).
I guess something interesting about the chapter is that a bunch of the theorems are if and only if which might imply that we could use some of the conditions to determine is p is a prime number. I know that is something that is difficult to do with very large numbers so is examining ring structures a helpful way to find primes? For example, if we took R/P for some number and found that it was an integral domain, then we could know that p is a prime (or irreducible if we are working with polynomials) by theorem 6.14, right? Or maybe I'm misinterpreting the theorems which would be a good thing to clear up.
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