# Maximal Cohen-Macaulay of finite injective dimension

Let $$(R,\frak m)$$ be a Cohen-Macaulay local ring. If there exists a maximal Cohen-Macaulay $$R$$-module $$M$$ with finite injective dimension, then can we conclude that $$R$$ has a canonical module?

Definition Let $$(R,\mathfrak{m},k)$$ be a commutative noetherian local ring with $$\text{dim}\,R=d$$. A finitely generated $$R$$ module $$G$$ is said to be Gorenstein of rank (or type) $$t$$ if $$\mu_{i}(\mathfrak{m},G)= \begin{cases} 0 \mbox{ if }i \neq d \\ t \mbox{ if }i = d \end{cases}$$
Clearly any Gorenstein module is a maximal Cohen-Macaulay $$R$$-module, as its depth is $$d$$, and has finite injective dimension (this is clear from Proposition 3.1.14 in Bruns-Herzog). From its definition, the canonical module is just a Gorenstein module of rank 1.