Algebraic Compactness OF $\prod M_\alpha / \oplus M_\alpha$

Abstract

In this note, we are working within the category $\rmod$ of (unitary, left) $R$-modules, where $R$ is a {\bf countable} ring. It is well known (see e.g. Kie{\l}pi\'nski & Simson [5], Theorem 2.2) that the latter condition implies that the (left) pure global dimension of $R$ is at most 1. Given an infinite index set $A$, and a family $M_\al\in\rmod$, $\al\in A$ we are concerned with the conditions as to when the $R$-module $$\prod/\coprod=\prod_{\al\in A}M_\al/\bigoplus_{\al\in A}M_\al$$ is or is not algebraically compact. There are a number of special results regarding this question and this note is meant to be an addition to and a generalization of the set of these results. Whether the module in the title is algebraically compact or not depends on the numbers of algebraically compact and non-compact modules among the components $M_\al$.