The set of smooth real functions $C^{\infty}=$ {$f: R\to R| \text{f is smooth}$} constitutes a vector space $C:$ {$C^{\infty}, +^C, •^S$} over the field {$R, +, •$}, where $$(v+^C w)(x):= v(x)+^R w(x)$$, and $$(a \in R •^S v)(x) := a•^R v(x)$$
But it seems like $C^{\infty}$ is also by itself a ring: Addition $+^C$ inherits the group properties from $R$. And defining an additional multiplication $C^{\infty} \times C^{\infty} \to C^{\infty}$ as follows $$(v •^C w)(x) := v(x) •^R w(x)$$ This multiplication constitutes a monoid on the set of smooth real functions. I am not sure if it is also a group on that set.
In any case, this set therefore constitutes a Ring (and possibly also a Field, I am not sure). Is this correct?
Are there generalizations of this? Are there many vector spaces that are also separately Fields or Rings (i.e. apart from the trivial way in which all fields are also vector spaces)?
Thanks for the answers so far. A lot of different concepts are falling into place.
A vector space is actually a module structure. In your case, yes it also has a ring structure. We called such structures as algebras. It can't be a field as $x^3$ is a smooth function but its inverse isn't.
Yes in the case of field extensions $L /K$, $L$ is also a field and has a basis over $K$. For example $\mathbb{Q}(\sqrt{2})$ over $\mathbb{Q}$.