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### A The Mathematical Idea behind **Modules**

As finite dimensional constructions in linear algebra over a field \(k\) boil down to solving (in)homogeneous linear systems over \(k\), the Gaussian algorithm makes the whole theory perfectly computable.

Hence, for homological algebra (viewed as linear algebra over general rings) to be computable one needs to find appropriate substitutes for the Gaussian algorithm, where finite dimensionality has to be replaced by finite generatedness.

Luckily such substitutes exist for many rings of interest. Beside the well-known Hermite normal form algorithm for principal ideal rings it turns out that appropriate generalizations of the classical GrÃ¶bner basis algorithm for polynomial rings provide the desired substitute for a wide class of commutative *and* noncommutative rings. Note that for noncommutative rings the above discussion has to be restricted to homological constructions leading to one-sided linear systems \(XA=B\) resp. \(AX=B\) (--> Principal limitation).