Example of a non-trivial ideal of ring of all continuous functions defined on all $\mathbb{R}$

Question

I'm trying to give an example of a non-trivial ideal of ring of all continuous functions defined on all $\mathbb{R}$.

But it seems that I don't have any good idea of how to construct this example. And I clearly don't understand the idea of rings of functions well (but it's pretty hard to explain which part exactly). I was searching for an example, but didn't find anything useful (maybe I missed something).

I think that when I understand how to solve this problem, it will help me understand this idea of ring of functions better.

Answer 1

Like, for instance, $7\Bbb Z$ is an ideal of $\Bbb Z$, so happen taking a function $f\ne 0$ and non-invertible of $C(\Bbb R)$. For example $I=\{x^2g(x) |g\in C(\Bbb R)\}$ is an ideal. Also the functions vanishing at a point $x_1$ or at a finite set $x_1,x_2,....,x_n$ form an ideal. (All this is easy to prove).

Answer 2

$A:=\mathcal{C}^0(\mathbb{R})$ is a commutative ring, hence it admits ideals, that is additive subgroups of $A$ which are stable under multiplication by any elements of $A$. Regarding your question, try to prove that the set of continuous functions which value at $0$ is $0$ is a non-trivial ideal of $A$.

Answer 3

As well as functions vanishing at one point, you could also take the ideal of functions vanishing at all but finitely many of the positive integers. This is a proper ideal, since it doesn't contain the constant function $1$ and it's non-empty, since it contains the truncated sine function (extended by a straight line to the left).