Gröbner bases and their applications over non-commutative rings

We plan the further development of two subsystems of Singular for computations with modules over non-commutative algebras. Plural works over the broad class of G-algebras, while Letterplace operates with objects in general finitely presented associative algebras. For both systems we will introduce rings like $\mathbb{Z}/m\mathbb{Z}$, $m \in \mathbb{N}_0,$ as coefficients and adopt Gröbner bases algorithms to this situation; algorithms for computing one- and two-sided syzygies and free resolutions will be developed (Letterplace) and enhanced (Plural). More specific algorithms for matrix theory and ring theory will be included.