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October 4th, 2012, 06:13 PM   #1
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Ideal Proof, possibly a trivial question

A is a ring and J is nonempty subset of A. Prove that if J is an ideal of A and , then J = A.

Does this imply that the the whole set can be J = {1}?

I know that for and for any we have that

So if , we have that (1)a = a = a(1) so . Since and does that mean J = A?

Please shine some light on this because I am struggling to understand it.
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October 4th, 2012, 07:13 PM   #2
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Re: Ideal Proof, possibly a trivial question

You're really overthinking this. You were done halfway through your proof.

If a is any element in the A then you just proved it's in J. So A is a subset of J. But we're already given that J is a subset of A. So A = J.
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October 4th, 2012, 08:08 PM   #3
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Re: Ideal Proof, possibly a trivial question

Thank you so much. Makes perfect sense now that you cleared that up and I am not going insane over this seemingly easy proof lol.
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