unique maximal semisimple submodule

Question

I read a property

An $R$-module $M$ has a unique maximal semisimple submodule.

I am not sure whether R as a ring needs to be commutative or not.

How to proof it?

Answer 1

The sum of semisimple modules is semisimple, so Zorn's lemma implies that a unique maximal semisimple module exists. (This is called the socle of $M$).

Answer 2

For any $R$ module $M$, one can form $\mathrm{soc}(M):=\sum\{N<M\mid N \text{ a simple submodule of } M\}$. ($R$ need not be commutative.)

This is called the socle of $M$. It is (by definition) semisimple, and you can easily see its maximality. It contains all simple submodules of $M$, therefore it contains all semisimple submodules of $M$.

Keep in mind though that this sum may be empty, that is, $M$ may not have any simple submodules at all, as is the case for the $\Bbb Z$ module $\Bbb Z_\Bbb Z$. In that case, we follow the usual convention about empty sums and conclude $\mathrm{soc}(M)=\{0\}$.