An elementary result on infinite and finite direct sums of modules

Abstract

Let $R$ be a ring, and consider a left $R$-module given with two (generally infinite) 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 finite subsets $I_0\subseteq I,$ $J_0\subseteq J,$ and a direct summand $Y\subseteq \bigoplus_{i\in I_0} C_i,$ such that $A \oplus Y \ =\ B \oplus(\bigoplus_{j\in J_0} D_j).$ We then note some ways that this result can and cannot be generalized, and some related questions.