What $2\mathbb Z/4\mathbb Z$ is not flat module in the ring $\mathbb Z/4\mathbb Z$

Question

Let $R=\mathbb Z/4\mathbb Z$ be a ring and the ideal $2\mathbb Z/4\mathbb Z$ is not flat module since $2\mathbb Z/4\mathbb Z$ is generated by a nilpotent element how to prove it?

Answer 1

You have an embedding $0\rightarrow 2Z/4Z\rightarrow Z/4Z$. A $Z/Z4Z$ bilinear form $b$ defined on $2Z/4X\times 2Z/4Z$ is zero. Since you will $b(2,2)=4b(1,1)$. This implies that the tensor product $2Z/4Z\otimes 2Z/4Z$ is zero.