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Sagar Kolte science forum beginner
Joined: 14 Apr 2005
Posts: 31

Posted: Sun Jul 02, 2006 2:25 pm Post subject:
Nature of a certian Ideal



consider the Ideal
I={set of all continous functions form R to R vanishing at 'a' , a fixed point }
in the ring of continous functions form R to R. with addition and multiplication being point wise.
R= set of real numbers
Is I principal?
Thank You. 

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William Elliot science forum Guru
Joined: 24 Mar 2005
Posts: 1906

Posted: Sun Jul 02, 2006 4:03 pm Post subject:
Re: Nature of a certian Ideal



On Sun, 2 Jul 2006, koltesagar wrote:
Quote:  consider the Ideal
I={set of all continous functions form R to R vanishing at 'a' , a fixed
point }

I = { f in C(R,R)  f(a) = 0 }
Quote:  in the ring of continous functions form R to R. with addition and
multiplication being point wise.
R= set of real numbers
Is I principal?
Some g with for all f, 
f(a) = 0 iff some h in C(R,R) with f = g*h ?
No. Notice g(a) = 0, sqr g in I.
some h with sqr g = g * h
h = (sqr g)/g is not continuous. 

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Virgil science forum Guru
Joined: 24 Mar 2005
Posts: 5536

Posted: Sun Jul 02, 2006 8:05 pm Post subject:
Re: Nature of a certian Ideal



In article
<8974198.1151853700195.JavaMail.jakarta@nitrogen.mathforum.org>,
koltesagar <sagar.kolte@gmail.com> wrote:
Quote:  consider the Ideal
I={set of all continous functions form R to R vanishing at 'a' , a fixed
point }
in the ring of continous functions form R to R. with addition and
multiplication being point wise.
R= set of real numbers
Is I principal?
Thank You.

One would have to have some fixed function f:R > R with f in I so that
for every g in I, g = h*f for some h in the ring.
But for every such g, one would have to have that g/f must have a finite
limit as x > a.
Consider g = real cube root of f,so g will be in I whenever f is in I.
What is lim_{x > a} g(x)/f(x) like? 

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David C. Ullrich science forum Guru
Joined: 28 Apr 2005
Posts: 2250

Posted: Mon Jul 03, 2006 8:24 am Post subject:
Re: Nature of a certian Ideal



On Sun, 02 Jul 2006 10:25:11 EDT, koltesagar <sagar.kolte@gmail.com>
wrote:
Quote:  consider the Ideal
I={set of all continous functions form R to R vanishing at 'a' , a fixed point }
in the ring of continous functions form R to R. with addition and multiplication being point wise.
R= set of real numbers
Is I principal?

No. If g is in I (and g is nonzero except at a) then there exists
f in I such that f/g > infinity at a, hence f is not in the ideal
generated by g.
************************
David C. Ullrich 

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