Let $f(t)\in\mathbb{C}[t]$, and let $I_f$ be the ideal in $\mathbb{C}[t]$ generated by $f(t)$.

The ideal $I_f$ has a natural $\mathbb{C}$-algebra structure. My question is the following:

For which polynomials $f\in \mathbb{C}[t]$ is the group of ($\mathbb{C}$-algebra) automorphisms of $I_f$ trivial?

Some examples:

$1)$ If $f(t)=t$, there are (of course) infinitely many automorphims, defined by $t\mapsto \alpha t$, for any $\alpha\in \mathbb{C}[t]-\{0\}$. This case is very simple to understand, since the ideal generated by $t$ is a free $\mathbb{C}$-algebra (without unit) and any homomorphism can be defined just by defining the image of $t$.

$2)$ If $f(t)=t(t-1)$ we were able to find only one automorphism, given by $f\mapsto f$, and $tf\mapsto f-tf$. It is an automorphims of order $2$.

One can observe that it is enough to define the above in $f$ and $tf$, since these two elements generate $I_f$, as a $\mathbb{C}$-algebra (although the algebra generated by these two polynomials are not free, so, one needs to be careful in defining homomorphisms)

$3)$ As a last example, we were not able to find any non-trivial automorphism of the ideal generated by $f(t)=t(t-1)(t-2)$.