  We prove that the property Add$(M)\subseteq$ Prod$(M)$ characterizes
$\Sigma$-algebraically compact modules if $|M|$ is not $\omega$-measurable.
Moreover, under a large cardinal assumption, we show that over any ring $R$
where $|R|$ is not $\omega$-measurable, any free module $M$ of
$\omega$-measurable rank satisfies Add$(M)\subseteq$ Prod$(M)$, hence the
assumption on $|M|$ cannot be dropped in general (e.g. over small non-right
perfect rings). In this way, we extend results from a recent paper by Simion
Breaz.
