Prove $2\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z} \overset{\sim}{=} \mathbb{Z}/2\mathbb{Z}$

Question

I need to prove that $2\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z} \overset{\sim}{=} \mathbb{Z}/2\mathbb{Z}$.

I know that $\mathbb{Z} \otimes_{\mathbb{Z}}\mathbb{Z}/2\mathbb{Z} \overset{\sim}{=} \mathbb{Z}/2\mathbb{Z}$, but using the universal property of the tensor prodcut doesn't seem to work here. Advice is greatly appreciated.

Answer 1

Hint:

As a $\mathbf Z$-module, $\;2\mathbf Z\simeq\mathbf Z$. Hence…