Subrings of a polynomial ring in one variable

Question

Suppose $k$ is a field and $k[x]$ the ring of polynomials in one variable with coefficients in $k$. Is there any classification theorem that tells us what a subring of $k[x]$ will look like? I can think of specific examples of subrings (such as the polynomials with $f'(0)=0$) but I know there are examples that show that a subring doesn't even have to be Noetherian, so I wonder what is known in general about a subring of $k[x]$.