In [[mathematics]], a '''module''' is one of the fundamental [[algebraic structure]]s used in [[abstract algebra]]. A '''module over a [[ring (mathematics)|ring]]'''<!--boldface per WP:R#PLA--> is a generalization of the notion of [[vector space]], overwherein athe [[Field (mathematics)|field]],whereinof [[scalar (mathematics)|scalars]] areis elementsreplaced ofby a given [[Ring (mathematics)|ring]]<!--Rings. inThe Wikipediaconcept haveof a''module'' 1,is accordingalso toa generalization of the conventionone inof [[Wikipedia:Manualabelian of Style/Mathematicsgroup]]-->, andsince anthe operationabelian calledgroups scalarare multiplicationexactly isthe definedmodules between elements ofover the ring and elements of the module. A module taking its scalars from a ring ''R'' is called an ''R''-module[[integer]]s.
Like a vector space, a module is an additive [[abelian group]], and scalar multiplication is [[Distributive property|distributive]] over the operation of addition between elements of the ring or module and is [[Semigroup action|compatible]] with the ring multiplication.
Modules are very closely related to the [[representation theory]] of [[group (mathematics)|group]]s. They are also one of the central notions of [[commutative algebra]] and [[homological algebra]], and are used widely in [[algebraic geometry]] and [[algebraic topology]].
