I'm studying flat morphisms of rings and of varieties and I'm stuck in a calculus at the end of an exercise. I'm working over an algebraically closed field $k$ and I want to show that the polynomial ring in one variable $k[t]$ is not flat over its subring $k[t^2-1,t(t^2-1)]$. I'm not really good in calculations with polynomials and I'm trying to figure out:
$1)$ Can we write $k[t^2-1,t(t^2-1)]$ in a much more useful way, in order to understand which type of polynomials are there?
$2)$ Given two ideals of the subring $k[t^2-1,t(t^2-1)]$, namely $I=(t^2-1)k[t^2-1,t(t^2-1)]$ and $J=t(t^2-1)k[t^2-1,t(t^2-1)]$, I need to show that the polynomial $p(t)=t(t^2-1)$ is NOT in $(I\cap J)k[t]$. At first, I thought that the intersection of the two ideals in the subring $k[t^2-1,t(t^2-1)]$ was the ideal generated by $t(t^2-1)^2$, but I'm not convinced. Since elements of the product of ideals $(I\cap J)k[t]$ are sum of products of elements of each of the ideals (and not just element of the type $fg$), I don't know how to prove that that specific polynomial $p(t)$ cannot be written as a sum of this type.
Any hint or advice would be appreciated!
Thank you!