Homological Conjectures

Question

Let

1. The strong Nakayama conjecture : If $M \in \rm{{mod\mbox{-}}}R$ and $\rm{Ext}^i(M,R)=0$ for $i \geq 0$, then $M$ is zero.
2. The generalized Nakayama conjecture
3. If $S$ is a simple module and $\rm{Ext}^i(S,R)=0$ for $i \geq 0$, then $S$ is zero.
4. The Auslander Reiten conjecture : If $M \in \rm{{mod\mbox{-}}}R$ and $\rm{Ext}^i(M,M \oplus R)=0$ for $i>0$, then $M$ is projective.
5. The Nakayama conjecture
6. The Tachikawa conjecture : If $M \in \rm{{mod\mbox{-}}}R$ and $\rm{Ext}^i(M,M)=0$ for $i>0$, then $M$ is projective.

I know that in Artin algebra there is the following relationship.

(1)$\Rightarrow$(2)$\Leftrightarrow$(3)$\Leftrightarrow$(4)$\Rightarrow$(5). In general ($R$ is a non commutative Noetherian ring), What is the relationship between them?

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