Invariant sets under linear operator and covering codes over modules

Let A be a finite commutative ring with identity. A subset H of the A-module An is called an R-short covering of An if every vector of this module can be written as a sum of a scalar multiple of an element in H and an A-linear combination with at most R canonical vectors. Let C(A,n,R) denote the smallest cardinality of an R-short covering of An. In order to explore the symmetries of such coverings, we investigate a characterization of invariant sets under certain linear operator over a finite ring A, based on its factorization into local rings. As a consequence, new classes of upper bounds on C(A,n,R) are obtained, extending previous results. Moreover, an optimal class on C(A,n,R) is derived from a construction of MDS codes over certain rings.