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October 4th, 2012, 06:13 PM  #1 
Member Joined: Oct 2009 Posts: 85 Thanks: 0  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. 
October 4th, 2012, 07:13 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,305 Thanks: 705  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. 
October 4th, 2012, 08:08 PM  #3 
Member Joined: Oct 2009 Posts: 85 Thanks: 0  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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