A covering problem over finite rings

Given a finite commutative ring with identity AA, define c(A,n,R)c(A,n,R) as the minimum cardinality of a subset HH of AnAn which satisfies the following property: every element in AnAn differs in at most RR coordinates from a multiple of an element in HH. In this work, we determine the numbers c(Zm,n,0)c(Zm,n,0) for all integers m≥2m≥2 and n≥1n≥1. We also prove the relation c(S×A,n,1)≤c(S,n−1,0)c(A,n,1)c(S×A,n,1)≤c(S,n−1,0)c(A,n,1), where S=FqS=Fq or ZqZq and qq is a prime power. As an application, an upper bound is obtained for c(Zpm,n,1), where pp is a prime.