Example of non-flat modules

Solution 1:

First of all, you should be more precise when denoting tensor products. Actually, in your framework, $\otimes$ may mean $\otimes_{\mathbb{C}}$ or $\otimes_R$.

If $\otimes=\otimes_{\mathbb{C}}$ then you are thinking at $R$ and $N$ as $\mathbb{C}$-vector spaces. Thus they are free and hence flat.

If $\otimes=\otimes_R$ then proceed as follows. Set $I:=\langle tx-t\rangle\subseteq R$. Consider the element $\left(x-1+I\right)\otimes_R t\in N\otimes_R \langle t\rangle$. Notice that since $1\notin \langle t\rangle$ we do not have $\left(x-1+I\right)\otimes_R t=\left(tx-t+I\right)\otimes_R 1=0$ in $N\otimes_R \langle t\rangle$. However, when you map this element to $N\otimes_R R$ via $N\otimes_R \langle t\rangle \to N\otimes_R R$ then now we have $1\in R$ and so $\left(x-1+I\right)\otimes_R t=\left(tx-t+I\right)\otimes_R 1=0$ in $R$.

Conclusion: the map is not injective and hence $N$ cannot be flat.