Editing Module (mathematics) (section)

== Types of modules ==
{{see also|Glossary of module theory}}

; Finitely generated: An ''R''-module ''M'' is [[finitely generated module|finitely generated]] if there exist finitely many elements ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> in ''M'' such that every element of ''M'' is a [[linear combination]] of those elements with coefficients from the ring ''R''.
; Cyclic: A module is called a [[cyclic module]] if it is generated by one element.
; Free: A [[free module|free ''R''-module]] is a module that has a basis, or equivalently, one that is isomorphic to a [[direct sum of modules|direct sum]] of copies of the ring ''R''. These are the modules that behave very much like vector spaces.
; Projective: [[Projective module]]s are [[direct summand]]s of free modules and share many of their desirable properties.
; Injective: [[Injective module]]s are defined dually to projective modules.
; 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 [[dual module|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</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 that are not simple (e.g. [[uniform module]]s).
; Faithful: A [[faithful module]] ''M'' is one where the action of each {{nowrap|''r'' ≠ 0}} in ''R'' on ''M'' is nontrivial (i.e. {{nowrap|''r'' ⋅ ''x'' ≠ 0}} for some ''x'' in ''M'').  Equivalently, the [[annihilator (ring theory)|annihilator]] of ''M'' is the [[zero ideal]].
; Torsion-free: A [[torsion-free module]] is a module over a ring such that 0 is the only element annihilated by a regular element (non [[zero-divisor]]) of the ring, equivalently {{nowrap|1=''rm'' = 0}} implies {{nowrap|1=''r'' = 0}} or {{nowrap|1=''m'' = 0}}.
; Noetherian: A [[Noetherian module]] is a module that satisfies the [[ascending chain condition]] on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps.  Equivalently, every submodule is finitely generated.
; Artinian: An [[Artinian module]] is a module that satisfies the [[descending chain condition]] on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
; Graded: A [[graded module]] is a module with a decomposition as a direct sum {{nowrap|1=''M'' = {{resize|140%|⨁}}<sub>''x''</sub> ''M''<sub>''x''</sub>}} over a [[graded ring]] {{nowrap|1=''R'' = {{resize|140%|⨁}}<sub>''x''</sub> ''R''<sub>''x''</sub>}} such that {{nowrap|''R''<sub>''x''</sub>''M''<sub>''y''</sub> ⊆ ''M''<sub>''x''+''y''</sub>}} for all ''x'' and ''y''.
; Uniform: A [[uniform module]] is a module in which all pairs of nonzero submodules have nonzero intersection.