As the title says, I am looking for a noetherian local ring $R$ of dimension 0 which is reduced (and thus Cohen-Macaulay) but not Gorenstein.
Due to Bruns, Herzog $-$ Cohen-Macaulay Rings Theorem 3.2.10 every noetherian local ring which is not Gorenstein fails to be Cohen-Macaulay or fails to be of type 1. Since every reduced ring of dimension $\leq 1$ is Cohen-Macaulay (see Stacks-Reference), we are thus looking for a noetherian, reduced local ring of dimension 0 that fails to be of type 1.
What constitutes a simple example of such a ring?
I am grateful for any kind of help or input! Cheers!
Best Answer
If you mean Krull dimension $0$, then I guess there is no example.
A reduced ring with Krull dimension $0$ is von Neumann regular, and a local VNR ring is a field.
The reducedness condition really kills things. $F_2[x,y]/(x,y)^2$ satisfies everything you said except it is not reduced.