Mittag-Leffler modules and definable subcategories. II

TL;DR

This paper extends the theory of Mittag-Leffler modules by analyzing definable subcategories and purity, providing new structural results and examples involving specific classes of rings and modules.

Contribution

It introduces new results on countably generated Mittag-Leffler modules within definable subcategories, correcting previous statements and exploring examples related to purity and specific ring classes.

Findings

Countably generated Mittag-Leffler modules are direct summands of preenvelopes of unions of pure chains.

The main result links module properties to the containing definable subcategory.

Examples include classes of divisible abelian groups and special rings like RD-rings and Warfield rings.

Abstract

In this note I take the opportunity to correct the last statement of Part I of same title and continue the study of uniform purity of epimorphisms in order to derive the main result, which states that--provided $R_R\in \langle\cal K\rangle$, equivalently, $\langle \cal L\rangle$ (the definable subcategory generated by $\cal L$) contains all absolutely pure left modules--every countably generated $\cal K$-Mittag-Leffler module in $\langle \cal L\rangle$ is a direct summand of a $\langle \cal L\rangle$-preenvelope of a union of an $\cal L$-pure $\omega$-chain of finitely presented modules. In conclusion I present a number of examples that starts with and grew out of the study of $\cal L$-purity (of monomorphisms in $\Bbb{Z}$-Mod) for $\cal L$, the definable subcategory of divisible abelian groups. Rings that get particular attention in this are RD-rings, Warfield rings and (the newly introduced) high rings.