Simple module: Difference between revisions

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In [[abstractmathematics]], more specifically [[modern algebra]] and [[module theory]], a (left or right) [[moduleModule (mathematics)|module]] ''SU'' over a [[ring (algebra)|ring]] ''R'' is calleda '''simple''' or module'''irreducible''', if itthere isare notno thenon-trivial [[zeroproper module]]submodules 0of and''U'' if(over its''R''). only [[submodule]]s are 0 andEquivalently, ''SU''. Understandingis thea simple modulesmodule over a''R'' ring[[if isand usuallyonly helpfulif]] becausethe these[[Cylic modulesmodule|cyclic formsubmodule]] thegenerated "buildingby blocks"every ofnon-zero allelement otheron modules''U'' inequals a certain sense''U''.

Simple modules, in some sense, form the "building blocks" for modules of [[Composition length|finite length]], analogous to the fact that finite [[simple group]]s form the building blocks for all finite groups. With this perspective, the understanding of simple modules is readily seen to be an important aspect of module theory.

An important advance in the theory of simple modules was the [[Jacobson density theorem]]. The Jacobson density theorem states: ''Let U be a simple right R-module and write D = End_R(U). Let A be any D-linear operator on U and let X be a finite D-linearly independent subset of U. Then there exists an element r of R such that x'''·'''A = x'''·'''r for all x in X.''<ref>Isaacs, Theorem 13.14, p. 185</ref>. In particular, any [[primitive ring]] may be viewed as (that is, isomorphic to) a ring of ''D''-linear operators on some ''D''-space.

A consequence of the Jacobson density theorem, is Wedderburn's theorem; namely that any right [[artinian ring|artinian]] [[simple ring]] is isomorphic to the matrix ring over a full [[division ring]]. This can also be established as a corollary of the [[Artin-Wedderburn theorem]].

In this article, all modules are assumed right [[Unital module|unital]] modules over some ring ''R'' (as opposed to "left module over ''R''").

== Examples ==