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On n-flat modules over a commutative ring

Let R be a commutative ring with unit, T an R-module, and n a positive integer. It is proved that T is n-flat over R if B⊗RT is B-torsionfree for each n–generated commutative R-algebra B. The converse holds if T is n–generated, in which case T is actually flat over R. Several other instances of the converse are established, but it is shown that the converse fails in general, even for R an integral domain, T an ideal of R, and n = 1.