Algorithms for determination of t-module structures on some extension groups

In \cite{kk04} the second and third author extended the methods of \cite{pr} and determined the \tm module structure on ${\text{Ext}}^1(\mathrm{\Phi },\mathrm{\Psi })$ where $\mathrm{\Phi }$ and $\mathrm{\Psi }$ were Anderson \tm modules over $A={\mathbf{F}}_q[t]$ of some specific types. This approach involved the concept of biderivation and certain reduction algorithm. In this paper we generalize the results of \cite{pr} and \cite{kk04} and present complete algorithm for computation of \tm module structure on ${\text{Ext}}^1(\mathrm{\Phi },\mathrm{\Psi })$ for \tm modules $\mathrm{\Phi }$ and $\mathrm{\Psi }$ such that $\text{rk}\mathrm{\Phi }>\text{rk}\mathrm{\Psi }\mathrm{.}$ The last condition is not sufficient for our algorithm to be executable. We show that it can be applied when the matrix at the biggest power of $\tau$ in ${\mathrm{\Phi }}_t$ is invertible. We also introduce a notion of $\tau$-composition series which we find suitable for the additive category of \tm modules and show that under certain assumptions on the composition series of $\mathrm{\Phi }$ and $\mathrm{\Psi }$ our algorithm is also executable.