Bases and closures under infinite sums

Motivated by work of Diestel and Kühn on the cycle spaces of infinite graphs we study the ramifications of allowing infinite sums in a module R^M. We show that every generating set in this setup contains a basis if the ground set M is countable, but not necessarily otherwise. Given a family N⊆R^M, we determine when the infinite-sum span N of N is closed under infinite sums, i.e. when N=N. We prove that this is the case if R is a field or a finite ring and each element of M lies in the support of only finitely many elements of N. This is, in a sense, best possible. We finally relate closures under infinite sums to topological closures in the product space R^M.