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Zbl 0865.16004
Albrecht, Ulrich; Facchini, Alberto
Mittag-Leffler modules and semi-hereditary rings. (English)
[J] Rend. Semin. Mat. Univ. Padova 95, 175-188 (1996). ISSN 0041-8994

A ring is right strongly non-singular or rsns if the finitely generated non-singular right modules are exactly the finitely generated submodules of free modules. Rsns right semi-hereditary rings are at the same time left semi-hereditary and the examples include Prüfer domains and the infinite direct product of the integers. A few equivalent conditions are given for semi-hereditary rsns rings to have finite right Goldie dimension; the infinite product of integers is not a ring of this kind. A right $R$-module $M$ is a Mittag-Leffler module if the natural map $M\otimes_R(\prod_{i\in I}M_i)\to\prod_{i\in I}(M\otimes_RM_i)$ is a monomorphism, for any collection $\{M_i:i\in I\}$ of left $R$-modules. This is equivalent to saying that every finite subset of $M$ is contained in a pure-projective pure submodule of $M$. Torsion free Mittag-Leffler abelian groups are exactly the $\aleph_1$-free groups, as shown by Azumaya and Facchini. This paper extends the latter result to rsns semi-hereditary rings as well as some results of Rothmaler on flat Mittag-Leffler modules over RD domains. A characterization of rsns right Goldie rings is given should they satisfy the right hereditary condition. For a maximal valuation ring or an almost maximal valuation ring, given are equivalent conditions to the Mittag-Leffler condition for any module and a torsion module respectively.
[R.Dimitrić (Berkeley)]

MSC 2000:: *16D80 Other classes of modules and ideals (assoc. rings and algebras)
13F05 Dedekind and Pruefer rings and their generalizations
16E60 Semihereditary and hereditary assoc. rings and algebras

Keywords: finitely generated non-singular right modules; finitely generated submodules of free modules; right semi-hereditary rings; Prüfer domains; right Goldie dimension; pure-projective pure submodules; Mittag-Leffler Abelian groups; rsns semi-hereditary rings; Mittag-Leffler modules; RD domains

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