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November 20th, 2012, 04:55 AM   #1
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maximal ideal of the ring Q -trivial question??

I am supposed to give an example or a counterexample:
Claim:
If is a ring homomorphism then takes maximal ideals of B to maximal ideals of A.

Solution:
If I let now I have an injective and surjective ring homomorphism ,and my question is why is a maximal ideal in ???
We know that which is NOT a maximal ideal in as f.ex giving a counterexample.
Maybe it is a trivial question but I just don't see it....Please help
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November 20th, 2012, 11:03 AM   #2
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Re: maximal ideal of the ring Q -trivial question??

We know that 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.
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November 21st, 2012, 03:16 AM   #3
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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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