C.M. integral domain which is not integrally closed

280 Views Asked by Bumbble Comm At 27 Mar 2026 - 7:50

If I am not wrong, it is known that:

{Regular rings} $\subsetneq$ {Complete intersection rings} $\subsetneq$ {Gorenstein rings} $\subsetneq$ {Cohen-Macaulay rings}.

It is known that a regular ring is normal (for example, this result appears in CRT of Matsumura).

I guess there exists a complete intersection/Gorenstein/Cohen-Macaulay ring (integral domain) which is not normal (integrally closed).

Can one please give an example of a non-local c.i./Gorenstein/C.M. integral domain which is not integrally closed?

Remark: From Serre's criterion for normality, an example I am looking for should NOT satisfy $R_1$.

There are 1 best solutions below

Bumbble Comm On 14 Jul 2015 - 9:38 BEST ANSWER

Any singular curve (irreducible and reduced) in the affine plane is clearly a complete intersection, but not normal.

For starters, take $K[x,y]/(x^2-y^3)$.