Let $a_1,\ldots,a_m \in \mathbb{C}[x_1,\ldots,x_n]$, with $m > n$, and let $R=\mathbb{C}[a_1,\ldots,a_m]$.
For example, $R=\mathbb{C}[x^2,x^3] \subseteq \mathbb{C}[x]$.
Question. Could we find all maximal ideals of $R$, in a similar way to Hilbert's Nullstellensatz for polynomial rings? Is it true that a maximal ideal of $R$ is of the following form: $\langle a_1-\lambda_1,\ldots,a_m-\lambda_m \rangle$, where $\lambda_1,\ldots,\lambda_m \in \mathbb{C}$?
If I am not wrong, such ideals are maximal (since the quotient is a field), but are all maximal ideals of that form? Perhaps I am missing something about the relations of the $a_i$'s?
Thank you very much!
$R$ is a quotient of $A=\mathbb{C}[y_1,y_2,\ldots,y_m]$ and maximal ideals of $A$ are of the form you require and then so are the maximal ideals of $R$.