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.


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.

  • $\begingroup$ Thanks very much, this seems to be a very satisfactory source! $\endgroup$ – Fred.Fred Feb 5 '14 at 9:53

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