$\Sigma$-algebraically compact modules and $\mathbf L_{\omega_1\omega}$-compact cardinals

TL;DR

This paper characterizes $ ext{Sigma}$-algebraically compact modules via the inclusion Add$(M) extsubseteq$ Prod$(M)$ and explores the influence of large cardinal assumptions on modules over rings with non-$ extomega$-measurable size.

Contribution

It establishes a new characterization of $ ext{Sigma}$-algebraically compact modules and extends prior results by incorporating large cardinal assumptions.

Findings

Add$(M) extsubseteq$ Prod$(M)$ characterizes $ ext{Sigma}$-algebraically compact modules when $|M|$ is not $ extomega$-measurable.

Under large cardinal assumptions, free modules of $ extomega$-measurable rank satisfy Add$(M) extsubseteq$ Prod$(M)$ over certain rings.

The results extend recent work by Breaz on modules and large cardinal hypotheses.

Abstract

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.