Simple module

In mathematics, a simple module (also called irreducible module ) is a special form of a module, ie, an algebraic structure. Simple modules fulfill a certain Minimalitätseigenschaft: They are " smallest" modules in the sense that they "contain" no even smaller modules. Simple modules are used in a sense as "building blocks" of other modules. Built on comparatively easy way of simple modules are for example semisimple modules or modules of finite length.

The concept of simplicity is also found in groups. There it is called analogy of simple groups. Similarly, one can define analogously a composition series of modules. Then apply similar results as for groups, especially the set of Jordan - Hölder.

Modules include as special cases of abelian groups and vector spaces. In these special cases, the simple modules are the simple abelian groups (ie, cyclic groups of prime order ) or one-dimensional vector spaces.

Definition

Be a ring and a module with.

Simply means that if and the only submodules of are.

Equivalent definitions

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

  • And every element except already generated
  • Is isomorphic to a quotient module, with a maximum ( left / right ) ideal of the ring.
  • Has length 1

Properties

Simple modules are always artinian and noetherian.

Many applications have the Lemma of Schur. This means for example, that the endomorphism ring of a simple module is a skew field.

Examples

  • Is a prime number, it is a simple module. This results from the fact that modules are particularly groups, and from the set of Lagrangian.
  • However, if not prime, so is not a simple module. Because then has a proper divisor, and of generated submodule is not yet the whole module.

( Summary: The simple -modules are precisely the primes for. )

  • Is a body that are so -modules nothing other than vector spaces over. These are simple if and only if they are one-dimensional.
  • Module (mathematics)
  • Algebra
  • Commutative Algebra
  • Group Theory
Maximal ideal Noetherian ring Schur's lemma Lagrange's theorem (group theory)
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