# 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 . ${\ displaystyle R}$${\ displaystyle M}$${\ displaystyle R}$${\ displaystyle M \ neq \ {0 \}}$

${\ displaystyle M}$simply means if and are the only sub-modules of . ${\ displaystyle \ {0 \}}$${\ displaystyle M}$${\ displaystyle M}$

### Equivalent Definitions

A module over a ring is simple if and only if it fulfills one of the following equivalent conditions: ${\ displaystyle M}$${\ displaystyle R}$

• ${\ displaystyle M \ neq \ {0 \}}$and each element except generated already${\ displaystyle 0}$ ${\ displaystyle M}$
• ${\ displaystyle M}$is isomorphic to a quotient module , where is a maximum (left / right) ideal of the ring .${\ displaystyle R / I}$${\ displaystyle I}$${\ displaystyle R}$
• ${\ displaystyle M}$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 . ${\ displaystyle End_ {R} (M)}$${\ displaystyle R}$${\ displaystyle M}$

## 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 .${\ displaystyle p}$${\ displaystyle \ mathbb {Z} / p \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$
• 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.${\ displaystyle n> 1}$ ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle n}$${\ displaystyle m}$${\ displaystyle m}$${\ displaystyle 0}$

(In summary: The simple modules are exactly those for prime numbers .) ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} / p \ mathbb {Z}}$${\ displaystyle p}$

• If there is a body , then modules are nothing but vector spaces over . These are simple if and only if they are one-dimensional.${\ displaystyle R}$${\ displaystyle R}$${\ displaystyle R}$