A covering problem over finite rings

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