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. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$