Galois Correspondence on Linear Codes over Finite Chain Rings

GivenS|R a finite Galois extension of finite chain rings andB anS-linear code we define twoGalois operators, the closure operator and the interior operator. We proof that a linear code is Galois invariant if and only if the row standard form of its generator matrix has all entries in the fixed ring by the Galois group and show a Galois correspondence in the class of S-linear codes. As applications some improvements of upper and lower bounds for the rank of the restriction and trace code are given and some applications to S-linear cyclic codes are shown.