Mittag-Leffler modules and definable subcategories. II

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.