Module (mathematics): Difference between revisions

Revision as of 18:17, 13 March 2023

Revision as of 07:01, 18 March 2023

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; Flat: A module is called [[flat module|flat]] if taking the [[tensor product of modules|tensor product]] of it with any [[exact sequence]] of ''R''-modules preserves exactness.

; Torsionless: A module is called [[torsionless module|torsionless]] if it embeds into its algebraic dual.

; Simple: A [[simple module]] ''S'' is a module that is not {0} and whose only submodules are {0} and ''S''. Simple modules are sometimes called ''irreducible''.<ref>Jacobson (1964), [https://books.google.com/books?id=KlMDjaJxZAkC&pg=PA4 p. 4], Def. 1; {{PlanetMath|urlname=IrreducibleModule|title=Irreducible Module}}</ref>

; Semisimple: A [[semisimple module]] is a direct sum (finite or not) of simple modules. Historically these modules are also called ''completely reducible''.

; Indecomposable: An [[indecomposable module]] is a non-zero module that cannot be written as a [[direct sum of modules|direct sum]] of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. [[uniform module]]s).