I don't quite understand theorem 3.8. Is it saying that for a unit, there are any number of equations that work with ax=b? ya=b is the example in the book and is that the only other case where an equation of this form works? or could wa=b and ca=b? I don't understand the multiplicative inverse example given in the book where 7^-1=3 and 3^-1=7 in Z10. What does the negative power mean for congruence classes? Lastly I was confused with the discussion on theorem 3.11 which shows that ax=1R for all a in R. But the proof didn't really make sense in my mind.
I likes the proofs of theorem 3.5 because they seemed like a lot of clever little tricks that I might have been able to think of (and it always feels nice to think you might be able to do the work the book did). I'm also happy to finally have subtraction in the toolkit because it is so helpful for proofs. I also thought it was very interesting that all fields are integral domains. But I guess I'm a little confused as to why only finite integral domains are fields. This whole topic is a little mind boggling to me and it's taking a little time for the terminology to sink in. Hopefully as we go, I'll get better at what a ring is instantly and then be able to build off of that better.
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