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I have written the lead, and expanded it to cover the Jacobson density theorem and the Artin-Wedderburn theorems |
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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 [[
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.
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