Corollary 4

Let A be an integral domain, K its field of fractions and M a torsion-free A-module such that M is canonically identified with a sub-A-module of $K \otimes_A M$. Then, for all $m \in \Omega$, $M_m$ is canonically identified with a sub-A-module of $K \otimes_A M$ and $M = \bigcap_{m \in \Omega} M_m$.

As M is identified with a submodule of $K \otimes_A M$, M, is identified with a sub-\&-module of $(K \otimes_A M)_m = K_m \otimes_A M$ (§ 2, no. 4, Theorem 1); as $K_m = K$, we see straightway that $M_m$ is torsion-free; moreover, the commutativity of the diagram

proves that the canonical mapping $M \to M_m$ is injective. The corollary then follows from Corollary 1 applied to the A-module $K \otimes_A M$ and its submodule M.

In particular, for every integral domain A,

(2)