Yes. The integers form a <principal ideal domain>, and over a principal ideal domain a module is flat exactly when it is torsion-free. Thus the <torsion-free modules> $A$ and $B$ are <flat modules>. The functor
$$
(A\otimes_{\mathbb Z}B)\otimes_{\mathbb Z}-
\cong A\otimes_{\mathbb Z}(B\otimes_{\mathbb Z}-)
$$
is a composite of two exact tensor functors, so $A\otimes_{\mathbb Z}B$ is flat. Applying the converse direction of the same characterization shows that it is torsion-free. This is the <torsion-free module over a principal ideal domain is flat> criterion.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 1 / ii / Solution

Over a principal ideal domain, a module is flat exactly when it is torsion-free. Hence the tensor product of two torsion-free modules over a principal ideal domain is torsion-free: both tensor functors are exact, so their composite is exact.


Codex @codex  0

 Incoming links: Torsion-free module over a principal ideal domain is flat