Ideals in a polynomial ring (2)

#ring_theory

Let $F$ be a field and $F [x]$ be the polynomial ring, which is a principal ideal domain. Let $R = {f \in F [x] : f^{'} \in (x)}$ , where $(x) \subset F [x]$ is the ideal generated by $x$ , and $f^{'}$ is the (formal) derivative of the polynomial $f$ . It is a fact that $R$ is a subring of $F [x]$ .

Prove that $x^{2}$ and $x^{3}$ are irreducible elements of $R$ .
Let $(x^{2}, x^{3})$ be the ideal in $R$ generated by $x^{2}$ and $x^{3}$ . Prove this is a proper ideal of $R$ .
Prove that $(x^{2}, x^{3})$ is not a principal ideal of $R$ .

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Let $F$ be a field and $F[x]$ be the polynomial ring, which is a principal ideal domain. Let $R=\{f\in F[x]:f'\in (x)\}$, where $(x)\subset F[x]$ is the ideal generated by $x$, and $f'$ is the (formal) derivative of the polynomial $f$. It is a fact that $R$ is a subring of $F[x]$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $x^2$ and $x^3$ are irreducible elements of $R$.
	\item Let $(x^2,x^3)$ be the ideal in $R$ generated by $x^2$ and $x^3$. Prove this is a proper ideal of $R$.
	\item Prove that $(x^2,x^3)$ is not a {\itshape principal} ideal of $R$.
\end{enumerate}