Simple module: Difference between revisions

Revision as of 10:52, 15 August 2009

Revision as of 10:53, 15 August 2009

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In [[mathematics]], more specifically [[modern algebra]] and [[module theory]], a (left or right) [[Module (mathematics)|module]] ''U'' over a ring ''R'' is a '''simple module''', if there are no non-trivial proper submodules of ''U'' (over ''R''). Equivalently, ''U'' is a simple module over ''R'' [[if and only if]] the [[Cyclic module|cyclic submodule]] generated by every non-zero element on ''U'' equals ''U''.

Simple modules, in some sense, form the "building blocks" for modules of [[Composition lengthseries|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.