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Let $C(\mathbb R)$ denote the ring of all continuous real-valued functions on $\mathbb R$, with the operations of pointwise addition and pointwise multiplication. Which of the following form an ideal in this ring?

a. The set of all $C^\infty$ functions with compact support.

b. The set of all continuous functions with compact support.

c. The set of all continuous functions which vanish at infinity, i.e. functions $f$ such that $\displaystyle\lim_{ x\to\infty} f(x) = 0$.

Please somebody help how can I solve this problem. I have no idea.

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    What have you tried? Which property or properties of an ideal are you having trouble checking, and for which option(s)?2012-09-18
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    i have tried from the definition of the ideal but can not reach to any coclusion for each option.any help or hints is welcome.thanks.2012-09-18
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    For a) If you add two $C^\infty$ functions with (possibly different) compact support, ist the sum still $C^\infty$? Is its support also compact? What about the negative of such a funciton? So, is (a) an additive subgroup of our ring? Then check: If xou multiply an element of (a) with a ring element, do you get an element of (a)? That is: If $f$ is $C^\infty$ with compact support and $g$ is just continuous, does $g\cdot f$ have compact support? And is it also $C^\infty$?2012-09-18

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