The triangular ring $T = \pmatrix{\mathbb{Z} & \mathbb{Q} \\ 0 & \mathbb{Q}}$ has the following properties:
• This generalizes to give an example of a ring with right global dimension $n$ and left global dimension $n+1$ by replacing $\mathbb{Z}$ by $R$, a commutative noetherian domain of global dimension $n$, then replacing $\mathbb{Q}$ by $K = Frac(R)$
• A similar example gives a ring which is noetherian but neither left nor right Ore. Just take $R = \pmatrix{S & 0 \\ S & I}$ where $S = \pmatrix{\mathbb{Z} & 0 \\ \mathbb{Z}_p & \mathbb{Z}_p}$ and $I = \pmatrix{\mathbb{Z} & 0 \\ 0 & \mathbb{Z}_p}$ is an $S$-ideal.