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I have a commutative algebra class and I heard the theorem from the professor:

Let $R$ be a Noetherian ring and $\{E_i : i\in I\}$ be a collection of injective $R$-modules then $\bigoplus_{i\in I} E_i$ is also injective.

My questions are:

  1. How to prove that? My professor does not talk about the proof of it (just states the theorem.) I asked to professor where I can find the proof and she asks that I might find the proof of it in some homological algebra textbook. However I can't find that.

  2. Does the following analogue of above theorem hold?

Let $R$ be an Artinian ring and $\{P_i : i\in I\}$ be a collection of projective $R$-modules then $\prod_{i\in I} P_i$ is also projective.

If $R$ is domain then $R$ is field so the theorem holds trivially. I wonder above statement holds for general (Artinian) ring.

Hanul Jeon
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1 Answers1

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The first statement is easily proved using Baer's criterion.

Consider a family $(E_i)_{i\in I}$ of injective modules, an ideal $J$ of $R$ and a homomorphism $f\colon J\to \bigoplus_{i\in I}E_i$. We want to show that this can be extended to a homomorphism $g\colon R\to\bigoplus_{i\in I}E_i$. Since $J$ and $f$ are arbitrary, this means that the direct sum is injective, by Baer's criterion.

Since $J$ is finitely generated, the image of $f$ is contained in a finite direct sum $\bigoplus_{i\in I'}E_i$, with $I'\subset I$ finite. Since finite direct sums of injective modules are injective, we can extend $f$ to a homomorphism $g'\colon R\to\bigoplus_{i\in I'}E_i$, and composing with the canonical embedding gives the thesis.

Note that this holds also for noncommutative rings (use right or left ideals, depending on the side $R$ is noetherian).

The dual result for artinian rings is more difficult to prove. You can find a proof in Anderson-Fuller, Ring and Categories of Modules, corollary 28.9. The proof is short, but it relies on several previous results, in particular Chase's theorem 19.20 and the fact that over perfect rings flat modules are projective. I don't see an easy way to prove it in the commutative case.

egreg
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