1. ## example, tensor product

Give an example of a ring $A$ and $A$-modules $B, C, D$ such that $0 \rightarrow B \rightarrow C$ is exact, yet

$0 \rightarrow B \otimes_A D \rightarrow C \otimes_A D$

is not exact.

I cannot think of an example where this would be true. Initially, I was thinking to use $A=\mathbb{Z}$ and then use ideals from that. However, I do not think that my example actually works. I need some help here. Thanks.

2. Originally Posted by pascal4542
Give an example of a ring $A$ and $A$-modules $B, C, D$ such that $0 \rightarrow B \rightarrow C$ is exact, yet

$0 \rightarrow B \otimes_A D \rightarrow C \otimes_A D$

is not exact.

I cannot think of an example where this would be true. Initially, I was thinking to use $A=\mathbb{Z}$ and then use ideals from that. However, I do not think that my example actually works. I need some help here. Thanks.
You can find the similar example of your problem in Example 3 of Dummit p 401.

You see that $0 \rightarrow \mathbb{Z} \overset{2}{\rightarrow} \mathbb{Z}$ is an exact sequence. The kernel of the second arrow is 0.
However, the induced map $0 \rightarrow \mathbb{Z} \otimes_{\mathbb{Z}}\frac{\mathbb{Q}}{\mathbb{Z}} \rightarrow \mathbb{Z}\otimes_{\mathbb{Z}} \frac{\mathbb{Q}}{\mathbb{Z}}$ is not exact since the kernel of the second map is not 0. Take $1 \otimes (1/2 + \mathbb{Z})$ as a counter example.