Is it possible to characterize all quotient rings of $\mathbb{C}[t]$?

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Let $I$ be a non-zero non-trivial ideal of $R:=\mathbb{C}[t]$. It is known that $I=(f)$, for some $f \in R$.

If $I$ is a prime or maximal ideal, then $\frac{R}{I}$ is an integral domain or a field, respectively.

If, for example $I=(t^2)$, then $\frac{R}{I}$ is not an integral domain.

Is there something interesting to say about $\frac{R}{I}$ in general?

By interesting I mean to find all isomorphism classes, for $I$ not prime.

Any comments and hints are welcome!