IJPAM: Volume 14, No. 1 (2004)

ALGEBRAIC COMPACTNESS OF $\prod M_{\al}/\bigoplus M_{\al}$

Radoslav M. Dimitric
Texas A&M University
P.O. Box 1675, Galveston, TX 77553, USA
e-mail: dimitric@tamug.edu

Abstract.In this note, we are working within the category $\rmod$ of (unitary, left) $R$-modules, where $R$ is a countable ring. It is well known (see e.g. Kie\lpinski and 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

\begin{displaymath}\prod/\coprod=\prod_{\al\in A}M_\al/\bigoplus_{\al\in A}M_\al\end{displaymath}

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 $M_\al$.

Received: April 5, 2004

AMS Subject Classification: 16D10, 16D80, 13C13

Key Words and Phrases: algebraically compact, product mod direct sum of modules, reduced product of modules, pure global dimension 1, countable rings

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 14
Issue: 1