Exercise 111.1.3. Suppose that $(A, {\mathfrak m}, k)$ is a Noetherian local ring. For any finite $A$-module $M$ define $r(M)$ to be the minimum number of generators of $M$ as an $A$-module. This number equals $\dim _ k M/{\mathfrak m} M = \dim _ k M \otimes _ A k$ by NAK.

Show that $r(M \otimes _ A N) = r(M)r(N)$.

Let $I\subset A $ be an ideal with $r(I) > 1$. Show that $r(I^2) < r(I)^2$.

Conclude that if every ideal in $A$ is a flat module, then $A$ is a PID (or a field).

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