# Which of the following form an ideal in this ring?

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

• The set of all $C^{\infty}$ functions with compact support.

• The set of all continuous functions with compact support.

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

• What are your thoughts on this? Commented Jan 20, 2014 at 2:38
• "Set of all $C^\infty$" is a little weird to say, but I guess you mean the subset of smooth functions $C^\infty\subset C(R)$? Commented Jan 20, 2014 at 13:42

By the definition of a subring, an ideal of $$C(\mathbb{R})$$ would be a subset $$I$$ of $$C(\mathbb{R})$$ such that:

• For $$f_1,f_2 \in I$$, $$f_1 + f_2 \in I$$
• For any $$f \in I$$ and any $$g \in C(\mathbb{R})$$, $$f\cdot g \in I$$.

With that in mind, the set of continuous functions with compact support is the only set you've described satisfying all of these properties.

## Why the other sets fail to be ideals:

### The set $$C^\infty(\mathbb{R})$$:

Note that $$f = 1$$ given by $$f(x) = 1$$ is in $$C^\infty(\mathbb{R})$$ and $$g$$ given by $$g(x) = |x|\in C(\mathbb{R})$$, but $$f\cdot g = g \not \in C^\infty(\mathbb{R})$$. Incidentally, this is why no proper subset of a ring containing the multiplicative identity can be an ideal of that ring.

### The set of functions with compact support:

Not all such functions are continuous, and so this fails to be a subset of $$C(\mathbb{R})$$ (I'll leave it to you to find a counterexample). Thus, it is not an ideal thereof.

### The set $$C_0(\mathbb{R})$$ of continuous functions vanishing at $$\pm \infty$$

Note that $$f$$ given by $$f(x) = e^{-|x|}$$ is in $$C_0(\mathbb{R})$$, and $$g$$ given by $$g(x) = e^{|x|}$$ is continuous. However, $$f \cdot g = 1$$ is not in $$C_0(\mathbb{R})$$.

• It would be quite useful to see why the others dont satisfy this definition Commented Jan 20, 2014 at 16:58
• how The set of all continuous functions with compact support becomes an ideal u did not explain...................... Commented Jan 17, 2017 at 18:53
• It seems that the question has changed since I posted my answer, so the first two non-ideals no longer apply. Commented Jan 17, 2017 at 20:33
• @sani you're right, and I don't intend to. If you have a question about it, perhaps you should post a new question. Commented Jan 17, 2017 at 20:34
• so what will be the answer? How u prove ? Commented Jan 17, 2017 at 20:38