Weak Injective and Weak Flat Modules

Let R be a ring. A left R-module M (resp., right R-module N) is called weak injective (resp., weak flat) if Ext(R)(1) (F,M) = 0 (resp., Tor(1)(R) (N,F) = 0) 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.