Exercise 20, § 8

Let $R$ be a principal ideal domain, $K$ its field of fractions. We denote by $B$ the subring of $\mathbf{M}_2(K)$ formed by the matrices $\begin{pmatrix} a & 0 \\ x & y \end{pmatrix}$ with $a \in R$, $x, y \in K$.

a) Now prove that $B$ is left noetherian but not right noetherian.

b) Now prove that $\mathrm{dh}(B) = 1$ and $\mathrm{dh}(B^\circ) = 2$.