Corollary 2

Let A be a not necessarily commutative ring, m the Jacobson radical of A and M a left A-module. Suppose that A/m is a field and that one of the following conditions holds: (i) M is finitely presented; (ii) m is nilpotent. Then the following properties are equivalent: (a) M is free; (b) M is projective; (c) M is flat; (d) the canonical homomorphism m \otimes_A M \to M is injective; (e) Tor_1^A(A/m, M) = 0.

The implications (a) \Rightarrow (b) \Rightarrow (c) \Rightarrow (d) are immediate. As A/m is a field, (A/m) \otimes_A M is a free (A/m)-module and Proposition 5 shows that (d) implies (a).

Finally, we know that Tor_1^A(A, M) = 0 and from the exact sequence 0 \to m \to A \to A/m \to 0 we therefore derive the exact sequence

this proves that Tor$_1^A(A/m, M)$ is isomorphic to the kernel of the canonical homomorphism $m \otimes_A M \to M$; whence the equivalence of (d) and (e). \*

It can be shown that, for every ring $A$ with Jacobson radical $m$ such that $A/m$ is a field, every projective $A$-module is free (Exercise 3).