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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 ...

Let $(R,m)$ be a Cohen-Macaulay local ring which possesses the canonical module $K_R$. Then $R$ is said to be an almost Gorenstein local ring, if there is an exact sequence $0 \to R \to K_R \to C \to ...