Sharp covering of a module by cyclic submodules

Let A be a finite commutative ring with identity. A subset H of the A  -module AnAn is called an R  -short covering of AnAn if every element of this module can be written as a sum of a multiple of an element in H and an A-linear combination with at most R   canonical vectors. Let c(A,n,R)c(A,n,R) be the minimum cardinality of an R  -short covering of AnAn. In this work, the numbers c(A,n,0)c(A,n,0) are computed when A is a direct product of chain rings (extending previous results by Yildiz et al.) and when A   is a finite local ring such that D(A)2={0}D(A)2={0}, where D(A)D(A) denotes the set of all zero divisors of A. In order to obtain these results, we develop a method based on action of group, the min–max principle and pairwise weakly linearly independent sets (a concept introduced in this article). A structural connection between classical covering and short covering is described too. Thus a combination of previous results is applied to improve known bounds on short coverings for several values.