---
tag: E5D4BD4193
kind: statement
role: theorem
name: Characterizations of semisimple modules
msc: [16D60]
uses: [09VDS2W1E8, 0RGZ10DFPG, G00C22AEDH, N3MRHDKDGC, 04RXTQA5ZA]
about: [N3MRHDKDGC]
sources:
  - url: https://en.wikipedia.org/wiki/Semisimple_module
---

## Statement

Let $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].
The following are equivalent:

1. $M$ is [[semisimple|N3MRHDKDGC]], that is, a direct sum of [[simple|G00C22AEDH]] submodules;
2. $M$ is a sum of simple submodules;
3. every [[submodule|0RGZ10DFPG]] $N$ of $M$ is a direct summand: there is a submodule $N'$ with $M = N \oplus N'$.

## Remarks

The step from 2 to 1, and to 3, uses [[Zorn's lemma|04RXTQA5ZA]].
Given $M = \sum_{i \in I} S_i$ and a submodule $N$, choose a set $J \subseteq I$ maximal such that the sum $N + \sum_{j \in J} S_j$ is direct.
Each $S_i$ meets this sum, and being simple it lies inside it, so the sum is all of $M$; taking $N = 0$ gives 1.
For 3 implies 2, one shows that every nonzero submodule contains a simple submodule, and that the sum of all simple submodules has a complement which must be zero.
Consequences: submodules and quotient modules of semisimple modules are semisimple.
Example: in a vector space every subspace has a complementary subspace, matching condition 3.

## Explanations

### L1 human

There are three ways to say that a module is built cleanly from simple pieces.
It is a direct sum of simple pieces; or it is merely covered by simple pieces, possibly overlapping; or every piece of it can be split off.
The theorem says these all mean the same thing.

### L2 human

Condition 3 is usually the easiest to check, and it is how Maschke's theorem proves semisimplicity of group representations.
Condition 2 shows quickly that quotients of semisimple modules are semisimple.
The equivalence is the starting point of the theory of semisimple rings, where every module satisfies these conditions.