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Flat modules over valuation rings. (English) Zbl 1123.13016

A number of ring and module properties are explored in stride; these properties include P-flatness, content flatness, local projectivity, finite projectivity and single projectivity, separable, projective and free modules. Most of these properties are discussed in the setting where the base ring \(R\) is a valuation ring. Some of the results are as follows:
Theorem 2. The ring \(R\) is left perfect if and only if each flat left module is a content module.
Theorem 7. Let \(R\) be a commutative ring with a self-FP-injective quotient ring \(Q\). Then each flat \(R\)-module is finitely projective iff \(Q\) is perfect.
Theorem 16 gives a characterization of commutative PP-Baer rings and singly projective rings.
Theorem 30. If \(R\) is a valuation ring, then the following three conditions are equivalent:
(1) \(R\) is maximal;
(2) each singly projective \(R\)-module is separable;
(3) each flat content module is locally projective.
Furthermore, characterizations are given for a valuation ring (with non-trivial zero divisors) to be strongly coherent and \(\pi\)-coherent.

MSC:

13F30 Valuation rings
16D40 Free, projective, and flat modules and ideals in associative algebras
13C11 Injective and flat modules and ideals in commutative rings
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