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Let $R$ be a noncommutative noetherian ring. Can I say that every indecomposable injective right module appears as a direct summand of a term in the minimal injective resolution of $R_R$? I know this is true for a noetherian commutative ring.

LSpice
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Zahra
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  • Can you give a reference for the noetherian commutative case? I can't even see why it's true for $R=\mathbb Z$ at this moment. – Peter Kropholler Dec 09 '22 at 21:16
  • See for example Theorem 1.1 put $R$ instead of $M$, in" Minimal injective resolutions with applications to dualizing modules and Gorenstein modules" a paper by for authors. – Zahra Dec 09 '22 at 21:25
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    I’m sure that Zahra doesn’t expect a positive answer, as for Artin algebras this is the famous (and open) Generalized Nakayama Conjecture. But maybe a counterexample is known for noetherian rings. – Jeremy Rickard Dec 10 '22 at 06:30
  • Yes, Jeremy. You are right like always. – Zahra Dec 10 '22 at 09:42

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