R is semisimple if and only if every R module is projective

Question

I am trying to show that $R$ is semisimple if and only if every R module is projective. What I know is that if $R$ is semisimple then every $R$ module $M$ is semisimple which implies $M$ can be expressed as a direct sum of simple modules i.e $R=\sum_{i\in I} Ra_i$ for some $a_i\in M$ then I am not sure how to use this to deduce $M$ is projective. I also know that $R$ is artinian but I don't know if this is relevant. There is already an answer to this question but it involved advanced techinques I am not familiar with.

Can anyone show me an easy way to see this

Thanks in advance

Zorns lemma is fine I was talking about global dimension or projective dimension — TheGeometer, Apr 27 '18 at 05:52
To show all modules are projective, you need that each submodule of a module has a complement. Given each module is a sum of simples, that's a fairly straightforward Zorn's lemma argument. — Angina Seng, Apr 27 '18 at 05:54

score 4 · Accepted Answer · edited Dec 13 '22 at 18:50

4

I guess you should know these characterisations of a semisimple $R$-module $M$:

$M$ is a sum of simple $R$-modules.
$M$ is a direct sum of simple $R$-modules.
For each submodule $N$ of $M$, $N$ admits a complement $M'$, i.e., there is a submodule $M'$ such that $M=N\oplus M'$.

Now back to your problem. $R$ is semisimple if and only if any (left or right) $R$-module $M$ is semisimple. This is equivalent to each $M$ being peojective, as there is always a free $R$-module $F$ such that $F\twoheadrightarrow M$ and $M$ having a complement means that the epimorphism admits a section $s\colon M\to F$.

edited Dec 13 '22 at 18:50

Aryaman Maithani

16,136

answered Apr 27 '18 at 06:15

josephz

2,294

Thank you very much for your response I guess we can also use the characteraisation that M is projective if and only is a direct summand of a free module and so since it has a complement in F we are done ? – TheGeometer Apr 27 '18 at 06:37
@TheGeometer Yeah, surely this way also works~ – josephz Apr 27 '18 at 06:48
1

I have made a small edit. – Aryaman Maithani Dec 13 '22 at 18:50

R is semisimple if and only if every R module is projective

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