1) The definition of an ideal on pg. 135 (second edition) is interesting - maybe it is not a commutative ring, but it is still possible for r*a to be in I as well as a*r. I guess what makes an ideal different than a subring? Does an ideal need to be a subring? If something is countable is it considered finite?
2) I would be interested in understanding better the usefulness of ideals; why are we interested in them?
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