Weak Injective and Weak Flat Modules

Abstract

Let R be a ring. A left R-module M (resp., right R-module N) is called weak injective (resp., weak flat) if (resp., ) for every super finitely presented left R-module F. By replacing finitely presented modules by super finitely presented modules, we may generalize many results of a homological nature from coherent rings to arbitrary rings. Some examples are given to show that weak injective (resp., weak flat) modules need not be FP-injective (resp., not flat) in general. In addition, we introduce and study the super finitely presented dimension (denote by l.sp.gldim(R)) of R that are defined in terms of only super finitely presented left R-modules. Some known results are extended.

Key Words:

• Super finitely presented dimension
• Super finitely presented module
• Weak flat module
• Weak flat preenvelope
• Weak injective module

2010 Mathematics Subject Classification:

• 16E30
• 16E10
• 16D50

ACKNOWLEDGMENTS

The first author would like to thank Professor Driss Bennis for his comments and help in writing this paper. The authors thank the referee for many considerable suggestions, which have greatly improved this paper.

Notes

Communicated by E. Kirkman.