When $\operatorname{id}R<∞$ then $\operatorname{id}R = \dim R$. The same holds for a finite free, projective or flat module instead of $R$, that is, $\operatorname{id}M = \dim R$. Does it hold for finite modules of finite $\operatorname{id}$? (Here $\operatorname{id}$ means injective dimension and $R$ is a Noetherian local ring.)
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As said in comments, this is equivalent to Bass' question, and proved in "On the ubiquity of Gorenstein rings". But the way to reach it is pretty long, specially regarding old terminology. I wonder if there is a direct proof for it? ( based on books like Matsumura's and Bruns_Herzog's)