$\biggl( \prod_{i=1}^{+ \infty} \mathbb{Z}/2^i \mathbb{Z} \biggr)\otimes_{\mathbb{Z}}\mathbb{Q} \neq 0$ without using $p$-adic integers

Question

I'm doing this exercise from Dummit-Foote:

Show that tensor products do not commute with direct products in general. [Consider the extension of scalars from $\mathbb{Z}$ to $\mathbb{Q}$ of the direct product of the modules $M_i = \mathbb{Z}/2^i \mathbb{Z}, \ i = 1 , 2, \ldots$ ]

Then we see that $$\prod_{i=1}^{+ \infty} \biggl( \mathbb{Z}/2^i \mathbb{Z} \otimes_{\mathbb{Z}}\mathbb{Q} \biggr) = \prod_i 0 = 0$$

But is there a way to conclude that $$\biggl( \prod_{i=1}^{+ \infty} \mathbb{Z}/2^i \mathbb{Z} \biggr)\otimes_{\mathbb{Z}}\mathbb{Q} \neq 0$$

without using $p$-adic integers ?

Answer 1

HINT:

You should be able to argue that $(1,1,1,\dots)\otimes 1\ne 0$, since no rational multiple of it can be $0$.