1) For the definition of a prime ideal, I thought that whenever bc was in P (the ideal) that b is in P or c is in P. I thought that was just a part of the absorption property of ideals? I guess it just means that it is possible for there to be a product of two things that are not originally in P, but the product is in P? Also, I don't fully understand why an ideal containing the 1 element is actually the entire ring. Is there any way we could go over that a little bit in class?
2) Maximals are a super interesting idea to me. But how could you ever really know that you couldn't get bigger than the ideal unless you had the entire ring?
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