Let $k$ be a field with the property that $k(t)$ (the rational function field in one variable over $k$ ) is isomorphic to $k$. (For instance, $\mathbb{Q}\left(x_1, x_2, \ldots\right)$ is such a field $k$.) Let $\theta: k(t) \rightarrow k$ be a fixed isomorphism. Let $p(t)$ be a fixed irreducible polynomial in $k[t]$, and let $A$ be the discrete valuation ring obtained by localizing $k[t]$ at the prime ideal $(p(t))$. On $R=A \oplus A$, define a multiplication by $(a, b)\left(a^{\prime}, b^{\prime}\right)=\left(a a^{\prime}, b a^{\prime}+\theta(a) b^{\prime}\right)$. Then $R$ is a ring with identity $(1,0)$. (1) Show that $R$ is a local ring with $\mathrm{rad} R=A \cdot p(t) \oplus A$. (2) Show that $\bigcap_{i=1}^{\infty}(\mathrm{rad} R)^i=(0) \oplus A$, and that this is the prime radical of $R$. (3) Show that every right ideal of $R$ is an ideal. (4) Show that $R$ is right noetherian but not left noetherian. (This exercise is to be contrasted with Krull's Theorem in commutative algebra which states that, for any commutative noetherian local ring $S, \bigcap_{i=1}^{\infty}(\operatorname{rad} S)^i=(0)$.)