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September 24th, 2012, 06:06 AM   #1
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inlusion of an ideal in the union of 2 ideals proof 2

I dont understand the following proof. Could someone explain it to me step by step.
Let be ideals
claim or
Proof (by contradition)
Suppose not. (meaning that if  , then and ??)
let (and why??)
and let (and why???)
then

Please enlighten me...
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September 25th, 2012, 09:43 AM   #2
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Re: inlusion of an ideal in the union of 2 ideals proof 2

Quote:
Originally Posted by rayman
I dont understand the following proof. Could someone explain it to me step by step.
Let be ideals
claim or
Proof (by contradition)
Suppose not. (meaning that if  , then and ??)
let (and why??)
and let (and why???)
then

Please enlighten me...

Proof: Suppose that . If we proceed by contradiction, we suppose that AND ; this is just the negation of the desired implication.

Since is not contained wholly within either or , this means that I can find some element that is in but NOT in (since ) (but it has to be in since ). Similarly for the other.

By properties of ideals, since , we must have . However, as , we have , and so , which is a contradiction. Thus we must have as we wanted-- that or .

It might help to actually draw some diagrams to see where each ideal is contained.
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September 27th, 2012, 03:53 AM   #3
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Re: inlusion of an ideal in the union of 2 ideals proof 2

thank you, that small reformulation helped me to get it.
Could you please give me any hits how to prove this claim?

Let be ideals and P be prime ideal.
Claim or
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September 27th, 2012, 04:41 AM   #4
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Re: inlusion of an ideal in the union of 2 ideals proof 2

Unless I'm reading this incorrectly, it should be very straightforward. If I is contained in P, then certainly a smaller ideal contained in I, such as , is also contained in P. (i.e., if , then we have .)

This doesn't use the fact that P is prime, so either I'm missing something or perhaps there's a typo? The only thing that might need to be shown is that is an ideal (which it is-- just show that it satisfies the definition of an ideal).
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September 27th, 2012, 04:49 AM   #5
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Re: inlusion of an ideal in the union of 2 ideals proof 2

yeah I thought about that too but it seemed so trivial so I thought there was some hidden trick.
My last claim is that
can I just prove it by noticing that ?
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September 27th, 2012, 05:03 AM   #6
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Re: inlusion of an ideal in the union of 2 ideals proof 2

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Originally Posted by rayman
My last claim is that
can I just prove it by noticing that ?

That seems to work to me! Just note that in general the product of two ideals is contained in the intersection of the ideals, and the claim is proven. It might be worthwhile showing this fact-- just pick an element in the product of the ideals and show it must be also in the intersection (not hard to show).
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September 27th, 2012, 05:09 AM   #7
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Re: inlusion of an ideal in the union of 2 ideals proof 2

that you very much for the help!
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