An elementary result on infinite and finite direct sums of modules

Abstract

Let R be a ring, and consider an R-module M given with two (generally infinite) direct sum decompositions, A⊕(⨁i∈ICi)=M=B⊕(⨁j∈JDj), such that the submodules A and B, and the Dj, are all finitely generated. We show that there then exist finite subsets I0⊆I, J0⊆J, and a direct summand Y⊆⨁i∈I0Ci, such that A⊕Y=B⊕(⨁j∈J0Dj).We then note some ways that this result can and cannot be generalized, and pose some related questions.