My Math Forum maximal ideal of the ring Q -trivial question??

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 November 20th, 2012, 04:55 AM #1 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 maximal ideal of the ring Q -trivial question?? I am supposed to give an example or a counterexample: Claim: If $\varphi: A\to B$ is a ring homomorphism then $\varphi^{-1}$ takes maximal ideals of B to maximal ideals of A. Solution: If I let now $\varphi: \mathbb{Z}\to \mathbb{Q}$ I have an injective and surjective ring homomorphism ,and my question is why is $M=(0)$ a maximal ideal in $\mathbb{Q}$??? We know that $\varphi^{-1}(0)=(0)$ which is NOT a maximal ideal in $\mathbb{Z}$ as f.ex $(0)\subset (2)\subset \mathbb{Z}$ giving a counterexample. Maybe it is a trivial question but I just don't see it....Please help
 November 20th, 2012, 11:03 AM #2 Senior Member   Joined: Aug 2010 Posts: 195 Thanks: 5 Re: maximal ideal of the ring Q -trivial question?? We know that $\mathbb{Q}$ is a field. In any field, any nonzero element is a unit, so any ideal with nonzero elements must have units, hence must have 1. Any ideal containing 1 is the whole ring, so any nonzero ideal of a field is the entire field. Therefore the (unique) maximal proper ideal of any field is the zero ideal.
 November 21st, 2012, 03:16 AM #3 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 Re: maximal ideal of the ring Q -trivial question?? Of course you are right I totally forgot about it. thank you

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# maximales ideal in q

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