homological algebra – finally generated module via PID with tensor with itself is zero


Hello, I have the next doubt about this problem:

Show that if $ A $ is a module finally created via a PID and $ A otimes _ { Lambda} A = 0 $, then $ A = 0 $.

I've done the next one, I'm looking at the next exact order

$ 0 rightarrow Tor (A) rightarrow A rightarrow A / Tor (A) rightarrow 0 $

We have that $ A / Goal (A) $ is therefore a finally created torsion-free module via a PID $ A / Goal (A) $ is a free module and that implies that the short exact sequence is split up.

That's why I have a morphism $ A / goal (A) right arrow A $ so that $ A / goal (A) right arrow A right arrow A / goal (A) $ is identity.

Now when I'm using tensor $ A $ I have the next composition

$ (A / Tor (A)) otimes A rightarrow 0 rightarrow (A / Tor (A)) otimes A $ is also identity

It follows $ (A / Tor (A)) otimes A = 0 $.

Since $ A / Tor (A) cong Lambda ^ {k} $ I have this $ A ^ {k} = 0 $

However, I don't know how to proceed and I got stuck with it, so any hint?