Let $n,k > 1$ be positive integers. Define the reduced polynomial rings :
$g^k_n = \Bbb R[X_n]/(G^k_n(X_n))$ where $G^k_n$ is a real polynomial of degree $n$ (that keeps the ring reduced). (k is just an index , not a power or such)
The question is how many nonisomorphic reduced polynomial rings of degree $n$ are there ? Lets call that $f(n)$.
With nonisomorphic I mean that the elements of the rings (between any 2 rings) are not linearly dependant.
For instance $ \Bbb R[X]/((X)^3-1)$ and $ \Bbb R[Y]/((Y)^3+1)$ are not linearly dependant and thus nonisomorphic. (one of those 2 rings is algebraicly closed , while the other is not hence they cannot be linearly dependant)
Is $f(n)=n$ ?
Im not sure if isomorphic is the correct terminology for what I want. If not, what is ? And is it still a kind of morphism ?
Thanks in advance.