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Please help me. I have read Finiteness of the Injective Hull by Alex Rosenberg and Daniel Zelinsky. An $R$-module $N$ has finite length if $N$ has a finite composition series. Alex Rosenberg and Daniel Zelinsky asked, "If $N$ has a finite length, does there exists an injective module containing $N$ which also has a finite length?".

In general topology, we have compact spaces, a generalization of finite spaces, and (Why is compactness so important?). This helps to understand the importance of compactness. My question is, "Why is finiteness so important for an injective hull?" Thank you for considering my question.

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    You read the paper, but did not find the author's motivation in the introduction persuasive? I don't think it's that finite composition length is important for injective hulls . As the author says "every module has an injective hull... if the module is finite length, then is the injective hull finite length?" Some things like this are inherited by the injective hull. For example, the uniform dimensions of $M$ and $E(M)$ are the same. – rschwieb May 03 '23 at 18:39
  • This problem was brought to the fore by Azumaya it says, while he was extending a notion to modules over an Artinian ring. Therefore, if you understand which modules have finite composition length hulls, then you might gain some benefit along the lines Azumaya was suggesting. – rschwieb May 03 '23 at 18:41

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