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Does anybody know an example of a Noetherian local ring $(R,m)$ which admits a maximal Cohen-Macaulay module of type one, but the ring $R$ itself is not CM?

If $C$ is a maximal CM module then the type of $C$, denoted by $r(C)$, is defined to be $\dim_{R/m}\mathrm{Ext}^d (R/m, C)$, where $d= \dim(R)$.

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There are many examples. Take $A$ be any regular local ring, and B is not a Cohen-Macaulay $A$-module. Now you can check that $R = A \ltimes B$ and $M = A$ is an example.

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The Following is my communications with Olgur Celikbas in researchgate:

Ogler:You may want to check: $R=k[[x,y]]/(x^2, xy)$ and $M=R/(x)$

me:does still such an emaple exist if we suppose that R is unmixed of dimension 2?(Unmixed = every associated prime is of height zero)

Ogler:Try this one: the ring of the example is from a paper of Costa, Huneke and Miller, titled "complete local domains of type 2 are CM".

$R=k[[x,y,z]]/(xz,yz)$. Then $dim(R)=2$, $depth(R)=1$ ($x-z$ is a non zero-divisor) and $R$ is reduced (so that associated primes are height zero.) Let $M=R/(z)$. Then $depth(M)=2$, i.e., $M$ is MCM. It looks like $Ext^0(k,M)=Ext^1(k,M)=0$ and $Ext^2(k,M)=k$ so $Type(M)=1$. (Type means the vector space dim of the first non zero $Ext$ w.r.t $R/m$)

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