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 Post subject: about the ring R=Z+XQ[X]Posted: Fri, 9 Mar 2012 04:38:24 UTC
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Joined: Fri, 12 Nov 2010 07:00:12 UTC
Posts: 48
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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 Post subject: Re: about the ring R=Z+XQ[X]Posted: Fri, 9 Mar 2012 04:41:31 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12075
Location: Austin, TX
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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 Post subject: Re: about the ring R=Z+XQ[X]Posted: Fri, 9 Mar 2012 04:50:18 UTC
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Joined: Fri, 12 Nov 2010 07:00:12 UTC
Posts: 48
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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 Post subject: Re: about the ring R=Z+XQ[X]Posted: Fri, 9 Mar 2012 05:09:47 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6004
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
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 so the ideal (X) is . The element X/2 is not in the ideal. Similarly ...

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 Post subject: Re: about the ring R=Z+XQ[X]Posted: Fri, 9 Mar 2012 05:14:15 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12075
Location: Austin, TX
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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 Post subject: Re: about the ring R=Z+XQ[X]Posted: Fri, 9 Mar 2012 05:14:43 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12075
Location: Austin, TX
outermeasure 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 so the ideal (X) is . 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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 Post subject: Re: about the ring R=Z+XQ[X]Posted: Fri, 9 Mar 2012 09:03:53 UTC
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Joined: Fri, 12 Nov 2010 07:00:12 UTC
Posts: 48
Thanks a lot, both moderators for the speedy reply!

I get it now.

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