Groebner bases over commutative rings and Applications to coding theory

We give a survey of results and applications relating to the theory of Gr\uf6bner bases of ideals and modules where the coefficient ring is a finite commutative ring. For applications, we specialize to the case of a finite chain ring.We discuss and compare the main algorithms that may be implemented to compute Gr\uf6bner and (in the case of a chain ring) Szekeres-like bases. We give an account of a number of decoding algorithms for alternant codes over commutative finite chain rings