MHB Proving Ideal Property of f(x)=0 for Every Rational x in $\mathcal{F}(\mathcal{R})$

[Kiwi1]

I am asked:

Prove that each of the following is an ideal of $\mathcal{F}(\mathcal{R})$:
a. The set of all f such that f(x)=0 for every rational x
b. The set of all f such that f(0)=0

My question is how do I know what the multiplicative operation is within the ring? Is multiplication the standard multiplication on real numbers or is it composition of functions?

I would expect it to be composition of functions but then if I choose g(x)=1 then I get g(f(x))=g(0) for any real x and this is not generally zero. So it must be the multiplication on reals.

Am I just supposed to know that?

 

[Fallen Angel]

Hi Kiwi,

I don't know where this exercises come from, but it must be specified what the product is.

It is usual to consider $\mathcal{F}(\mathbb{R})$ the ring of real valued functions with pointwise product, and with this product your statements are true (they aren't with composition).

Try to prove it with pointwise multiplication and let us know if you encounter any other problem. ;)