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yoyobarn
 Post subject: about the ring R=Z+XQ[X]
PostPosted: Fri, 9 Mar 2012 04:38:24 UTC 
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Hi, for the ring R=Z+XQ[X], consider the
ascending chain of principal ideals (X)<(1/2 X)<(1/4 X)<... which is known to not terminate.

May I ask what element is found in (1/2 X) that cannot be found in (X)? (so as to get the proper inclusion)


Thanks a lot.
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Shadow
 Post subject: Re: about the ring R=Z+XQ[X]
PostPosted: Fri, 9 Mar 2012 04:41:31 UTC 
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yoyobarn wrote:
Hi, for the ring R=Z+XQ[X], consider the
ascending chain of principal ideals (X)<(1/2 X)<(1/4 X)<... which is known to not terminate.

May I ask what element is found in (1/2 X) that cannot be found in (X)? (so as to get the proper inclusion)


Thanks a lot.


Uhhh, Q is a field, so (1/2 x)=(x).

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yoyobarn
 Post subject: Re: about the ring R=Z+XQ[X]
PostPosted: Fri, 9 Mar 2012 04:50:18 UTC 
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but according to this website:
http://www.lohar.com/researchpdf/exampl ... n_play.pdf

(pg 352)
It was stated that (X)<(1/2 X)<(1/4 X)<... was a infinite chain of proper inclusions that do not terminate?

Thanks for the help.
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outermeasure
 Post subject: Re: about the ring R=Z+XQ[X]
PostPosted: Fri, 9 Mar 2012 05:09:47 UTC 
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Shadow wrote:
yoyobarn wrote:
Hi, for the ring R=Z+XQ[X], consider the
ascending chain of principal ideals (X)<(1/2 X)<(1/4 X)<... which is known to not terminate.

May I ask what element is found in (1/2 X) that cannot be found in (X)? (so as to get the proper inclusion)


Thanks a lot.


Uhhh, Q is a field, so (1/2 x)=(x).


Be careful! The ring is \mathbb{Z}+X\mathbb{Q}[X] so the ideal (X) is \mathbb{Z}X+X^2\mathbb{Q}[X]. The element X/2 is not in the ideal. Similarly ...

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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Shadow
 Post subject: Re: about the ring R=Z+XQ[X]
PostPosted: Fri, 9 Mar 2012 05:14:15 UTC 
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yoyobarn wrote:
but according to this website:
http://www.lohar.com/researchpdf/exampl ... n_play.pdf

(pg 352)
It was stated that (X)<(1/2 X)<(1/4 X)<... was a infinite chain of proper inclusions that do not terminate?

Thanks for the help.


Oh, I missed the X in front. Then yes, it works. The element 1/2X is not in (X).

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Shadow
 Post subject: Re: about the ring R=Z+XQ[X]
PostPosted: Fri, 9 Mar 2012 05:14:43 UTC 
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outermeasure wrote:
Shadow wrote:
yoyobarn wrote:
Hi, for the ring R=Z+XQ[X], consider the
ascending chain of principal ideals (X)<(1/2 X)<(1/4 X)<... which is known to not terminate.

May I ask what element is found in (1/2 X) that cannot be found in (X)? (so as to get the proper inclusion)


Thanks a lot.


Uhhh, Q is a field, so (1/2 x)=(x).


Be careful! The ring is \mathbb{Z}+X\mathbb{Q}[X] so the ideal (X) is \mathbb{Z}X+X^2\mathbb{Q}[X]. The element X/2 is not in the ideal. Similarly ...


Yeah, I noticed that after I posted it, my new post should address it.

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yoyobarn
 Post subject: Re: about the ring R=Z+XQ[X]
PostPosted: Fri, 9 Mar 2012 09:03:53 UTC 
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Thanks a lot, both moderators for the speedy reply!

I get it now.
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