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A note on flat modules. (English) Zbl 0718.16001

Let R be a ring with 1. For a right ideal I of R and submodule L of a free left R-module \(R^ n\), we say L is finitely I-presented if there is an R-epimorphism f: \(R^ p\to L\) such that the inverse image \(f^{- 1}(I^ m)\) of the subgroup \(I^ m\cap L\) of L is of the form \(K+I^ p\), where K is a finitely generated submodule of ker f. Define \({\mathcal M}(I)\) to be the class of all right R-modules that have a set of generators annihilated by I. The following conditions are equivalent: (1) R/I is flat, and if \(\{M_{\alpha}|\alpha\in A\}\subseteq {\mathcal M}(I)\), then \(\{\prod M_{\alpha}|\alpha\in A\}\) is flat; (2) for any set A, \((R/I)^ A\) is flat; (3) every finitely generated submodule of a free left R-module of finite rank is finitely I-presented; (4) every finitely generated left ideal of R is finitely I-presented. When \(I\subseteq J\) and R/I flat, six conditions equivalent to R/J being a flat right R-module are given.

MSC:

16D40 Free, projective, and flat modules and ideals in associative algebras
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
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