Test modules for flatness

Abstract

A right $R$-module $M$ is said to be a test module for flatness (shortly: an f-test module) provided for each left $R$-module $N$, Tor$(M,N)=0$ implies $N$ is flat. f-test modules are flat version of the Whitehead test modules for injectivity defined by Trlifaj. In this paper the properties of f-test modules are investigated and are used to characterize various families of rings. The structure of a ring over which every (finitely generated) right $R$-module is flat or f-test is investigated. Abelian groups that are Whitehead test modules for injectivity or f-test are characterized.