Abstract
A right R-module P R is called flat if P ⊗ R — is an exact functor on \( {}_R\mathfrak{M} \), the category of left R-modules. Specifically, this requires that, if A → B is injective in \( {}_R\mathfrak{M} \), then P ⊗ R A → P ⊗ R B is injective also. Projective modules are flat, but flat modules enjoy an important property not shared by projective modules: they are closed w.r.t. direct limits. In particular, a module is flat if all f.g. submodules are flat. Over ℤ, the flat modules are just the torsion-free abelian groups.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Lam, T.Y. (2007). Flat Modules and Homological Dimensions. In: Exercises in Modules and Rings. Problem Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-48899-8_2
Download citation
DOI: https://doi.org/10.1007/978-0-387-48899-8_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98850-4
Online ISBN: 978-0-387-48899-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)