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Keywords:
countably generated relative Mittag-Leffler; atomic; pure-projective module; definable subcategory; divisible module over RD-domains; preenvelope; coherent; Noetherian; v.N. regular; RD-purity
countably generated relative Mittag-Leffler; atomic; pure-projective module; definable subcategory; divisible module over RD-domains; preenvelope; coherent; Noetherian; v.N. regular; RD-purity
Summary:
A countably generated module is Mittag-Leffler if and only if it is pure-projective, i.e., a direct summand of a direct sum of finitely presented modules. Trying to generalize this description to countably generated relative Mittag-Leffler modules, one runs into serious obstacles. The last theorem of Part I of the same title describes them as what was called uniform relative pure epimorphic images of a specific kind of $\omega $-limits of finitely presented modules. A second part of this theorem attempted to make the $\omega $-limits in question more concrete. In the formulation the application of those uniform pure epimorphisms was erroneously omitted. Correcting this led to a better result to be presented here. It states that under a mild assumption on the context, every countably generated relative Mittag-Leffler module `in the context' is a direct summand of a certain preenvelope of a union of a relatively pure $\omega $-chain of finitely presented modules. In conclusion a number of examples are presented that start with and grew out of the study of $\cal L$-purity of monomorphisms in $\mathbb {Z}$-Mod for $\cal L$, the definable subcategory of divisible Abelian groups. Rings that get particular attention in this are RD-rings.
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