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I am a student, prepping for a final exam in graduate Abstract Algebra.

My professor has told me that he will be giving us the following two problems in class to turn in:

(1) Given that R is an integral domain with "said condition", show that if this 'said condition' is true, then R is field if and only if "this is true"

(2) Given that R is a commutative ring with $1 \ne 0$, prove that "some property involving a unit" is true. Or if it is a proper ideal, then it is a non-unit.

I can bring one page of theorems/notes in class to solve these two proposed questions.

My question to everyone is this: If you were presented with these hints towards a final exam, what would you put in your notes to help you solve these questions? What is your thinking process on why it is needed?

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For this level of question, about the only things I could advise you to take notes on is what all the terms mean.

1) is hopelessly generic, and it could be any number of things. The best preparation would probably just be to solve as many homework problems from your algebra book as possible in the sections you learn about commutative rings, domains and fields.

2) is sort of guessable. I guess it is "Show that (x)=R iff x is a unit."

Again, I think that the best preparation for such tests is to solve as many related exercises as possible to gain experience with the things. In this way you'll bring far more to the test than you could probably bring on notes.

I'm not saying notes are useless: they're valuable as a study strategy, or simply if something is very complex. But like I said the level of those questions suggests you would benefit the most by mastering the vocabulary and as many basic exercises as possible.

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