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.
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$\begingroup$ 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. $\endgroup$– Peter KrophollerDec 9, 2022 at 21:16
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$\begingroup$ 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. $\endgroup$– ZahraDec 9, 2022 at 21:25
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2$\begingroup$ 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. $\endgroup$– Jeremy RickardDec 10, 2022 at 6:30
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$\begingroup$ Yes, Jeremy. You are right like always. $\endgroup$– ZahraDec 10, 2022 at 9:42
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