Covering classes, strongly flat modules, and completions

  • Alberto FacchiniEmail author
  • Zahra Nazemian


We study some closely interrelated notions of Homological Algebra: (1) We define a topology on modules over a not-necessarily commutative ring R that coincides with the R-topology defined by Matlis when R is commutative. (2) We consider the class \( \mathcal {SF}\) of strongly flat modules when R is a right Ore domain with classical right quotient ring Q. Strongly flat modules are flat. The completion of R in its R-topology is a strongly flat R-module. (3) We prove some results related to the question whether \( \mathcal {SF}\) a covering class implies \( \mathcal {SF}\) closed under direct limits. This is a particular case of the so-called Enochs’ Conjecture (whether covering classes are closed under direct limits). Some of our results concern right chain domains. For instance, we show that if the class of strongly flat modules over a right chain domain R is covering, then R is right invariant. In this case, flat R-modules are strongly flat.


Covering class Strongly flat module Completion Cotorsion module R-topology 

Mathematics Subject Classification

Primary 16E30 16W80 Secondary 18G15 



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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PadovaPaduaItaly
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

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