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Let $R$ be a left noetherian ring, then it is well known that Matlis proved that any injective $R$-module decomposes into a sum of indecomposable injectives, each of form $E(R/I)$ for some irreducible left ideal $I$. If moreover $R$ is commutative, we know much more: indecomposable injectives correspond to prime ideals via $P \rightarrow E(R/P)$ and each $E(R/P)$ is filtered by vector spaces over quotient field of $R/P$, which makes it quite analogical to injective abelian groups in my eyes.

What is known in the non-commutative case? Is there also some nice filtration as in the commutative case?

Thanks in advance for any reference.

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The noncommutative analogue of this result is explained in Bo Stenstrom's Rings of Quotients, particularly in Chapter 7, “Hereditary Torsion Theories for Noetherian Rings”.

It's been a long time since I looked at this, so I forget the details, but my impression was that the noncommutative theory was not nearly as nice.

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  • $\begingroup$ Thanks very much, this seems to be a very satisfactory source! $\endgroup$
    – Fred.Fred
    Commented Feb 5, 2014 at 9:53

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