Let $R=\frac{k[X_1,\ldots,X_n]}{I}$, where $k$ is a field and $I$ is an ideal. Let $M$ be a finitely generated module over $R$. I would like to compute a minimal generating set for $M$. As $R$ is not a polynomial ring, Quillen-Suslin cannot be applied here. However I would like to know whether there are any special cases in which mingen can be computed, algorithmically?
In particular, I am more interested in the following situation:
$I$ is a principal ideal. By some other calculations, I know that $M$ is generated by $r$ generators, which satisfy 1 relation. i.e. $syz(M)$ is a cyclic module. In the language of computational algebra system, $M$ is a right module on $r$ generators satisfying 1 relation.
By some other theory I know that $M$ is minimally generated by $r-1$ generators. Is there any way to compute these $r-1$ generators?
Thanks in advance.