I know that [if $R$ is Noetherian local with a finite module of finite injective dimension, then *$R$ is Cohen-Macaulay*](https://math.stackexchange.com/questions/710644/does-operatornameid-m-dim-r-hold-for-finite-modules-of-finite-injective-d). 



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> Can one add assumptions on $M$, so that $R$ be *Gorenstein* or *Complete intersection* or *Regular*? 
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thank you.



if $R$ is Noetherian local with a finite module of finite injective dimension and if "?" , then $R$ is "Gorenstein"
If you admit $M$ cyclic as additional assumtion, then $R$ is Gorenstein by a theorem in Peskine-Szpiro paper "Dimension projective finie et cohomologie locale", Theorem II.5.5.


