So when you work over a commutative ring, this result is quite well known. I am wondering if the same holds true for an arbitrary ring; that is, if $R$ is some (possibly noncommutative) ring, does the following implication hold:
$$\text{Flat module} \implies \text{Torsion-free}\ ?$$
In particular, I am considering a ring which has no zero divisors (i.e. a domain).
If I add in the condition that the module is finitely generated, can I also claim the reverse implication?
Best Answer
Pretty much the same proof as in the commutative case shows that flat implies torsion-free for modules over a noncommutative ring $R$:
Let $a\in R$, not a right zero divisor. Then the map $\mu_a:R\to R$ given by $r\mapsto ra$ is an injective left $R$-module homomorphism. If $M$ is a flat right $R$-module, then $\operatorname{id}_M\otimes\mu_a:M\otimes_RR\to M\otimes_RR$ must also be injective. But this is isomorphic to the map $M\to M$ sending $m\in M$ to $ma$. So $a$ does not annihilate any non-zero element of $M$.