Show that $(T^2, T^3)$ is not a principal ideal in $\{ a + T^2 f\mid a \in \mathbb{Q}, f \in \mathbb{Q}[T] \}$.

Question

Let $R = \{ a + T^2 f \mid a\in \mathbb{Q}, f \in \mathbb{Q}[T] \}$. Show that the ideal $(T^2, T^3)$ is not principal in $R$.

So far, I've shown that the the set of invertible elements of $R$ is equal to the set of invertible elements of $\mathbb{Q}$ and that the elements $T^2$ and $T^3$ are irreducible but not prime in $R$.

Answer 1

Suppose the contrary: $(T^2,T^3)=(g)$ for some $g\in R$. Then $g\mid T^2$ and $g\mid T^3$ in $R$, and therefore in $\mathbb Q[T]$. It follows that $g\in\{1,T,T^2\}$. The second and the third case are not possible since $g\in R$ and $T\notin R$. Thus $g=1$. We have $(T^2,T^3)=(1)$, so $1=T^2h_1+T^3h_2$ with $h_1,h_2\in R$, false.