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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)? –  Qiaochu Yuan Sep 18 '12 at 17:50
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. –  mintu Sep 18 '12 at 17:56
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 $C^\infty$? –  Hagen von Eitzen Sep 18 '12 at 18:11
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1 Answer

First check that they’re all closed under the ring operations; after all, if they’re not, they can’t be ideals. Assuming that they pass that test, you have to check that they have the absorbing property of an ideal. If you take a function in the ideal and multiply it pointwise by a continuous real-valued function, do you always get a function in the ideal? For (c), for instance, consider the function $$f(x)=\frac1{x^2+1}\;;$$ it’s clearly in the ideal. What happens when you multiply it by the function $g(c)=x^2+1$? For (b): if $f$ has compact support, and $g$ is continuous, does $fg$ always have compact support? Hint: $\operatorname{supp}fg=\operatorname{supp}f\cap\operatorname{supp}g$. For (a): Is the product of a $C^\infty$ function with a continuous function always $C^\infty$? Hint: Absolute value.

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