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

TL;DR

This paper investigates the algebraic compactness of certain quotient modules formed from products and direct sums of modules over countable rings, generalizing previous results and linking compactness to the properties of component modules.

Contribution

It provides new criteria for algebraic compactness of product-over-sum modules in the context of countable rings, extending existing results in the literature.

Findings

Algebraic compactness depends on the number of compact modules among components.

Generalizes previous results on algebraic compactness of product modules.

Connects module properties to the structure of their component families.

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 $M_\al$.