Flat modules over valuation rings |
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Authors: | F Couchot |
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Institution: | Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR 6139, Département de mathématiques et mécanique, 14032 Caen Cedex, France |
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Abstract: | Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self-injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent.A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self-FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect. |
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Keywords: | Primary 13F30 13C11 secondary 16D40 |
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