[Math] Set of continuous functions as a ring

ring-theory

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?

Best Answer

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?

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