# Simple module

In mathematics , a **simple module** (also called an **irreducible module** ) is a special form of a module , i.e. an algebraic structure . Simple modules fulfill a certain minimality characteristic: They are "smallest" modules in the sense that they do not "contain" any smaller modules. In a certain sense, simple modules serve as “building blocks” for other modules. Semi-simple modules or modules of finite length are, for example, constructed in a comparatively easy manner from simple modules .

The concept of simplicity can also be found in groups . There one speaks analogously of simple groups . Similarly, you can define a composition series for modules . Similar results then apply as for groups, especially the Jordan-Hölder theorem .

As special cases, modules include Abelian groups and vector spaces . In these special cases the simple modules are the simple Abelian groups (ie cyclic groups with prime order ) or one-dimensional vector spaces.

## definition

Be one ring and one - module with .

simply means if and are the only sub-modules of .

### Equivalent Definitions

A module over a ring is simple if and only if it fulfills one of the following equivalent conditions:

- and each element except generated already
- is isomorphic to a quotient module , where is a maximum (left / right) ideal of the ring .
- has the length 1.

## properties

Simple modules are always Artinian and Noetherian .

Schur's lemma has many applications . This says, for example, that the endomorphism ring of a simple module is an oblique body .

## Examples

- If a prime number , then it is a simple module. This follows from the fact that modules are in particular groups and from Lagrange's theorem .
- On the other hand , if there is
*no*prime number, there is no simple module. Because then it has a real divisor , and the sub- module generated by is neither nor the whole module.

(In summary: The simple modules are exactly those for prime numbers .)

- If there is a body , then modules are nothing but vector spaces over . These are simple if and only if they are one-dimensional.