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, $A\oplus ({⨁}_{i\in I}{C}_i)=M=B\oplus ({⨁}_{j\in J}{D}_j),$ 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 $Y\subseteq {⨁}_{i\in I_0}{C}_i,$ such that $A\oplus Y=B\oplus ({⨁}_{j\in J_0}{D}_j).$ We then note some ways that this result can and cannot be generalized, and some related questions.

arXiv comment4 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy. Revision has simpler version of formula (1.2), and other, minor, improvements in wording etc