Difficult:
In this section of 3.1, I had a hard time understanding why you would have to have 4 new axioms to determine whether or not the subset of a ring would be a subset. If its a subset of a ring, I don't understand why these 4 axioms need to be re-fulfilled. And if these 4 do, why don't the other 4? I don't understand why closed under addition and multiplication, and a 0 element is in the subset, and that there is a solution to a + x = 0 ( the 0 element in the subset). I didn't understand why these are the 4 axioms that need to be fulfilled.
Reflection:
I really like using the Cartesian product of z modulo 6 and z modulo 4 to see if it is a ring. I like being able to write out addition and multiplication tables to organize the information to help determine if the Cartesian product is a ring. I also like that we can redefine addition and multiplication to find other rings and use the axioms to see if the set is a ring or if that ring has any subrings.
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