Set of continuous functions as a ring

Question

In Artin there is a question to test whether the set is a ring or not - $S$ = {Set of all real valued continuous functions}

(f+g)(x) = f(x)+g(x)

And

(f.g)(x) = f(g(x))

I have proved this as a ring -

• $S$ is closed under Addition and Multiplication(which is composition in this case)
• Additive inverse is in $S$
• Additive identity $a(x)=0$ is in $S$
• Multiplicative identity is identity map $a(x) = x$ is in $S$

Please correct me if there is something wrong?

Answer 1

It seems that you are checking whether it is a subring. However, ask yourself first, subring of what? If you have a ring $R$ and a subsets $S$ of it, you can easily check if $S$ is a subring just by checking a few things. But to prove that something is a ring without happily seeing it as a subset of some ring with the same operations, you have to check all the ring axioms. Remember that the addition and multiplication in a ring have to be compatible. Does that hold in your case above?

Answer 2

Suppose $f(x)=x^2$ and $g(x)=1$.

Then $f(g(x)+g(x))=4$ but $f(g(x))+f(g(x))=2$.

So this is not distributive.