Small's Example from noncommutative algebra...
The triangular ring $T = \pmatrix{\mathbb{Z} & \mathbb{Q} \\ 0 & \mathbb{Q}}$ has the following properties:
- It's right noetherian but not left noetherian
- It's right hereditary but not left hereditary
- The right global dimension is 1 but the left global dimension is 2
- 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.
Having been trained to think in a commutative world, I found the existence of an example for any one of these to be surprising. The fact that they were all (basically) the same example is even more amazing.
