Hi, nice write-up. It might be worth while to point out the difference between an irreducible element and prime element. In the ring Z[sqrt -5] I would not call 2 and 3 prime numbers but instead irreducible numbers. Call a number irreducible if its only factorization includes only units and itself. There is a good reason why mathematicians often define prime numbers as follows: p is prime if whenever p|ab then p|a or p|b. This generalizes nicely to ideals. We can say an ideal A divides an ideal B, if there is an ideal C such that B=AC, written A|B. Then prime ideal can be defined the same way as a prime number, and this is nice because we get to see more of an overlap now. P is a prime ideal if whenever P|AB, then P|A or P|B. It turns out this definition is equivalent to your definition for prime ideal (I think it is always equivalent, though I could be wrong in some cases).
My point is not to say that you are wrong in any way, I just thought you might enjoy the response.

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