An elementary result on infinite and finite direct sums of modules

Let R be a ring, and consider an R-module M given with two (generally infinite) direct sum decompositions, A & REG; (& REG;i & ISIN;I Ci) = M = B & REG; (& REG;j & ISIN;J Dj), such that the submodules A and B, and the Dj, are all finitely generated. We show that there then exist finite subsets I0 C I, J0 C J, and a direct summand Y C & REG;i & ISIN;I0 Ci, such that A & REG;Y = B & REG;(& REG;j & ISIN;J0 Dj). We then note some ways that this result can and cannot be generalized, and pose some related questions.& COPY; 2023 Elsevier Inc. All rights reserved.