**Definition**: a (not necessarily commutative) left and right Noetherian ring $R$ is said to be Auslander-Gorenstein if

(i) $R$ has finite left and right injective dimension (in which case it turns out they are equal)

(ii) For any f.g. $R$-module $M$ and submodule $N$ of $Ext^j(M, R)$, $Ext^i(N, R) = 0$ whenever $i < j$

There are two results which relate this to Krull dimension:

If $R$ is commutative Noetherian and has finite injective dimension $n$ then (a) it is Auslander-Gorenstein and (b) $Kdim(R) = n$.

**[1]**If $R$ is Auslander-Gorenstein of injective dimension $n$ then $Kdim(R) \leq n$.

**[2]**

So the question is: does there exist a (left and right) Noetherian ring of finite injective dimension $n$ with $Kdim(R) > n$? That is, does the second result actually require $R$ to be Auslander-Gorenstein, or did it only need the injective-dimension-finite part?

Obviously, those two results mean that such an $R$ cannot be commutative or Auslander-Gorenstein. Unfortunately, the commonly quoted example of a Noetherian ring which is not Auslander-Gorenstein - $\left( \begin{array}{cc} k & V \\ 0 & k \end{array} \right)$ for $V$ a $k$-vector space of dimension at least 2 - is Artinian, so doesn't help.

**[1]**: Hyman Bass. On the ubiquity of Gorenstein rings, 1963.

**[2]**: K. Ajitabh, S. P. Smith, and J. J Zhang. Auslander-Gorenstein rings, 1998.