My Math Forum Ideal Proof, possibly a trivial question

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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 $1 \in J$, then J = A. Does this imply that the the whole set can be J = {1}? I know that for $1 \in J$and for any $a \in A$ we have that $1a,a1 \in J$ So if $1 \in J$, we have that (1)a = a = a(1) so $a \in J$. Since $a \in J$ and $a \in A$ 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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