An elementary result on infinite and finite direct sums of modules

TL;DR

This paper proves a fundamental result about the structure of modules with two infinite direct sum decompositions, showing how finite parts can be aligned under certain conditions.

Contribution

It establishes a new elementary theorem relating finite submodules within infinite direct sum decompositions of modules.

Findings

Existence of finite subsets aligning direct sum decompositions

Conditions under which finite parts can be matched

Discussion on generalizations and related questions

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.