I am studying the ring homomorphism of the following function:
If $f: M \to N$ is a smooth function, then $$f^*: C^{\infty}(N) \to C^{\infty}(M), \textbf{ defined by } \phi \mapsto \phi \circ f$$ is a ring homomorphism.
I know that the given function $f^*$ should satisfy that for every $h,g \in C^{\infty}(M),$ we have that $f^*(hg) = f^*(g) f^*(h).$ my question is should the operation between $g,h$ be multiplication or composition and why?
Could someone explain this to me please?
It is multiplication. If $h$ and $g$ are maps from $N$ into $\Bbb R$ how could you possibly compose them? Only if $N=\Bbb R$.
When I write that it is multiplication, what I mean is that $hg$ is the map $n\mapsto h(n)g(n)$