# “Direct sums of injective modules over Noetherian ring is injective” and its analogue

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.

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).