Hi Let $R = K[X_1,\ldots, X_n]$ where $K$ is a computable field. Suppose we are given two modules with presentations
$$ R^n \rightarrow R^m \rightarrow M \rightarrow 0 $$ and $$ R^l \rightarrow R^p \rightarrow N \rightarrow 0 $$
Then is it possible to verify whether $M$ is isomorphic to $N$ (using a computer algebra software)?
Longtime ago (in 2003) this was not possible. I do not know whether it is possible now.