How do we show that something is not a flat module?

Question

Now a days I'm reading some basic module theory.Here is one question on which I'm stuck:

Let $R=k[t]$ and $S= \dfrac {k[x,t]} { (x) \cap (x,t)^2} $ Show that $S$ is not a flat $R$-module.

I'm using the following definition of flat module.

Def: A flat module over a ring $R$ is an $R$ module $M$ such that taking the tenor product over $R$ with $M$ preserves exact sequences.

So,to solve this question we need to find a exact sequence which does not preserve the taking tenor product,but I'm unable in finding such a sequence and also I don't know much properties of flat modules So any ideas to find such exact sequence?

Answer 1

Luckily for us $k[t]$ is a PID, so that flat is equivalent to torsion-free. Also, $$I:=(x)\cap (x,t)^2=(x)\cap (x^2,xt,t^2)=(x^2,xt).$$ Now $x$ is torsion in $S$, because it is killed by $t$.

Notice that in general flat implies torsion-free, but not viceversa. Consider for instance the ideal $(x,t)$ as a $k[x,t]$-module.