My Math Forum inlusion of an ideal in the union of 2 ideals proof 2

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 September 24th, 2012, 06:06 AM #1 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 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 $I, J_{1}, J_{2}$ be ideals claim $I\subset J_{1}\cup J_{2}\Rightarrow I\subset J_{1}$ or $I\subset J_{2}$ Proof (by contradition) Suppose not. (meaning that if $I\subset J_{1}\cup J_{2}$ , then $I\not\subset J_{2}$ and $I\not\subset J_{1}$??) let $x\in I\Rightarrow x\not\in J_{2}$ (and $x\in J_{1}$ why??) and let $y\in I\Rightarrow y\not\in J_{1}$(and $y\in J_{2}$why???) then $\begin{cases}x+y\not\in J_{1}\\ x+y\not\in J_{2}\end{cases}$ Please enlighten me...
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 $I, J_{1}, J_{2}$ be ideals claim $I\subset J_{1}\cup J_{2}\Rightarrow I\subset J_{1}$ or $I\subset J_{2}$ Proof (by contradition) Suppose not. (meaning that if $I\subset J_{1}\cup J_{2}$ , then $I\not\subset J_{2}$ and $I\not\subset J_{1}$??) let $x\in I\Rightarrow x\not\in J_{2}$ (and $x\in J_{1}$ why??) and let $y\in I\Rightarrow y\not\in J_{1}$(and $y\in J_{2}$why???) then $\begin{cases}x+y\not\in J_{1}\\ x+y\not\in J_{2}\end{cases}$ Please enlighten me...

Proof: Suppose that $I\subseteq J_1\cup J_2$. If we proceed by contradiction, we suppose that $I\not\subset J_1$ AND $I\not\subset J_2$; this is just the negation of the desired implication.

Since $I$ is not contained wholly within either $J_1$ or $J_2$, this means that I can find some element $x$ that is in $I$ but NOT in $J_2$ (since $I\not\subset J_2$) (but it has to be in $J_1$ since $I\subseteq J_1\cup J_2$). Similarly for the other.

By properties of ideals, since $x,y\in I$, we must have $x+y\in I$. However, as $x\not\in J_2, y\not\in J_1$, we have $x+y\not\in J_1, x+y\not\in J_2$, and so $x+y\not\in I$, which is a contradiction. Thus we must have as we wanted-- that $I\subseteq J_1$ or $I\subseteq J_2$.

It might help to actually draw some diagrams to see where each ideal is contained.

 September 27th, 2012, 03:53 AM #3 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 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$I,J,P$ be ideals and P be prime ideal. Claim $I\subset P$ or $J\subset P \Rightarrow I\cap J\subset P$
 September 27th, 2012, 04:41 AM #4 Senior Member   Joined: Nov 2011 Posts: 100 Thanks: 0 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 $I \cap J$, is also contained in P. (i.e., if $I\subseteq P$, then we have $I\cap J \subseteq I \subseteq P$.) 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 $I \cap J$ is an ideal (which it is-- just show that it satisfies the definition of an ideal).
 September 27th, 2012, 04:49 AM #5 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 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 $I\cap J\subset P\Rightarrow IJ\subset P$ can I just prove it by noticing that $IJ\subset I\cap J$?
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 $I\cap J\subset P\Rightarrow IJ\subset P$ can I just prove it by noticing that $IJ\subset I\cap J$?

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).

 September 27th, 2012, 05:09 AM #7 Senior Member   Joined: Sep 2010 From: Germany Posts: 153 Thanks: 0 Re: inlusion of an ideal in the union of 2 ideals proof 2 that you very much for the help!

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# union of two ideals is not an ideal

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