Weakly-τ-Flat Modules

In this paper, we introduce and study a torsion-theoretic generalization of the weakly-flat module using the concept of τ -closed submodule for an idempotent radical τ . We introduce a new class of modules called weakly-τ -flat. We study these modules, generalizing several results both on extending modules and flat modules. We study generalizations and characterizations of IF ring, (finitely) Σ -CS ring, and C-ring by weakly-τ -flat modules.