Fix a positive integer $n$ and consider the polynomial ring $R=\mathbb{Q}[x_1,\ldots,x_n]$, which is Noetherian. Given a description of an ideal in $R$, it often takes some work to find a finite generating set, although this can sometimes be done algorithmically. For example, fix $m$, and for each $n$-tuple $f_1,\ldots,f_n\in S:=\mathbb{Q}[y_1,\ldots,y_m]$, consider the ideal $I_{f_1,\ldots,f_n}$ which is the kernel of the map $R\to S$, $x_i\mapsto f_i$. A finite generate set for $I_{f_1,\ldots,f_n}$ can be computed algorithmically from $f_1,\ldots,f_n$ using a Gröbner basis.
Are there classes of ideals for which there is no algorithm whose input is an ideal in the class and whose output is a finite generating set for that ideal?
I think I am asking the following: does there exist a recursive set $S\subset \mathbb{N}\times R$ such that for each $n\in \mathbb{N}$, $\{r\in R:(n,r)\in S\}$ is an ideal of $R$, with the property that there is no recursive set $S'\subset S$ such that for each $n\in \mathbb{N}$, $\{r\in R:(n,r)\in S'\}$ is a finite generating set for $\{r\in R:(n,r)\in S\}$?