The elements of $ (\mathbb {Z}/2\mathbb{Z})⊗_{\mathbb {Z}}\mathbb {Z}$

Question

consider the tensor product of $ \mathbb {Z}$-modules $\mathbb {Z}$ and $\mathbb {Z}/2\mathbb{Z} $ . Why $\bar {1}⊗1\neq 0$ in $\mathbb {Z}/2\mathbb{Z}⊗_{\mathbb {Z}}\mathbb {Z}$.

Answer 1

Well in a tensor product $M\otimes_A N$ of two $A$-modules, a simple tensor $m\otimes n \in M\otimes N$ is zero if and only if for all $A$-module $P$ and for all $b \in Bil_A(M\times N,P)$, $b(m,n)=0$. In your case it shouldn't be to hard to find a convenient bilinear map to $\mathbb{Z}/2\mathbb{Z}$.

Answer 2

By the canonical $\mathbb{Z}$ module isomorphism you have that $\mathbb {Z}/2\mathbb{Z}⊗_{\mathbb {Z}}\mathbb {Z}$ is isomorphic to $\mathbb {Z}/2$, given by the isomorphism that sends $a⊗b$ to ab. Therefore, $1⊗1$ gets sent to 1 which is not zero in $\mathbb{Z}/2$.