The Covering Problem for Chinese Rings

Given a direct sum G of cyclic groups, we find a sharp bound for the minimal number of proper subgroups whose union is G. This generalizes to sums of cyclic modules over more general rings, and we are able to solve this covering problem for all local, Artinian, or Dedekind rings (and even for monoids). We also minimally cover all finitely generated non-cyclic modules over quasi-local rings. These and related questions have been the subject of much past study, but only for finite groups. Our methods differ from those found in the literature, and make it possible to work in the general setting of Chinese rings, which generalize the Chinese Remainder Theorem (and include the above cases). We also address a related problem: given a finite set M that is a direct sum of cyclic R-modules, and a distinguished element m ∈ M , what is the smallest number of cosets of proper R-submodules whose union is M \ {m}? When R = Z, this question is related to various conjectures in the literature.