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Let $R$ be a ring, and consider a left $R$-module given with two (generallyinfinite) direct sum decompositions, $A\oplus(\bigoplus_{i\in I}C_i)=M=B\oplus(\bigoplus_{j\in J} D_j),$ such that the submodules $A$ and $B$and the $D_j$ are each finitely generated. We show that there then exist finitesubsets $I_0\subseteq I,$ $J_0\subseteq J,$ and a direct summand $Y\subseteq\bigoplus_{i\in I_0} C_i,$ such that $A \oplus Y \ =\ B \oplus(\bigoplus_{j\inJ_0} D_j).$  We then note some ways that this result can and cannot be generalized, andsome related questions.

An elementary result on infinite and finite direct sums of modules