Examples of Noetherian local rings which are not Gorenstein

Question

Can anyone give me an example of a Noetherian local ring which is not a Gorenstein ring?

Answer 1

Every Gorenstein local ring is Cohen-Macaulay and of type $1$ [see Proposition 3.2.10 of Bruns and Herzog Cohen-Macaulay Rings]. So if, (1) a ring is not CM or (2) the type of a ring is not $1$, then that ring will not be Gorenstein. Therefore you have plenty of examples, such as:
$k[x,y,z]/(x) \cap (y,z)$,
$k[x,y,u,v]/(x,y)\cap (u,v)$,
2.1.18 in Bruns and Herzog,
2.1.19 in Bruns and Herzog.
(which are not CM).

AND

you can use the type to reach other examples; such as Matt E's example. Also example 3.2.11(a) of Bruns and Herzog have other examples.

Answer 2

Sure. The ring $k[x,y]/(x^2,xy,y^2)$ (here $k$ is a field) is not Gorenstein.

Answer 3

The ring $k[[x,y]]/(x^2 ,xy,y^2 )$ (here $k$ is a field) is not Gorenstein, because the zero ideal is not irreducible.