Let $K$ be an algebraically closed field and consider the $K$-algebra $K[t]$ i.e the $K$-algebra of all polynomials in the indeterminate $t$ with coefficients in $K$.
Now consider an ideal $I$ of $K[t]$ then since $K[t]$ is PID we have that $I= \langle p(t) \rangle$ for some polynomial $p(t) \in K[t]$.
Suppose now that we have the situation that $K[t]/I \cong A$ where $A$ is any connected $K$-algebra of dimension $3$. (by connected algebra I mean that is $A$ is not the direct product of two algebras).Here $\cong$ means isomorphism as $K$-algebras.
Question: Can we always guarantee that $K[t]/I$ is isomorphic to $K[t]/(t^{3})$ as $K$-algebra? or that $K[t]/(t^{3})$ is isomorphic to $K^{3}?$ are these all the possible cases? or what can we say about the ideal $I$?