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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.
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.
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?
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 non-zero 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.

 Quote: Thank You.

************************

David C. Ullrich

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