Editing Module (mathematics) (section)

=== Generalizations ===
A ring ''R'' corresponds to a [[preadditive category]] '''R''' with a single [[object (category theory)|object]]. With this understanding, a left ''R''-module is just a covariant [[additive functor]] from '''R''' to the [[category of abelian groups|category '''Ab''' of abelian groups]], and right ''R''-modules are contravariant additive functors. This suggests that, if '''C''' is any preadditive category, a covariant additive functor from '''C''' to '''Ab''' should be considered a generalized left module over '''C'''. These functors form a [[functor category]] '''C'''-'''Mod''', which is the natural generalization of the module category ''R''-'''Mod'''.

Modules over ''commutative'' rings can be generalized in a different direction: take a [[ringed space]] (''X'', O<sub>''X''</sub>) and consider the [[sheaf (mathematics)|sheaves]] of O<sub>''X''</sub>-modules (see [[sheaf of modules]]). These form a category O<sub>''X''</sub>-'''Mod''', and play an important role in modern [[algebraic geometry]]. If ''X'' has only a single point, then this is a module category in the old sense over the commutative ring O<sub>''X''</sub>(''X'').

One can also consider modules over a [[semiring]]. Modules over rings are abelian groups, but modules over semirings are only [[commutative]] [[monoid]]s. Most applications of modules are still possible. In particular, for any [[semiring]] ''S'', the matrices over ''S'' form a semiring over which the tuples of elements from ''S'' are a module (in this generalized sense only). This allows a further generalization of the concept of [[vector space]] incorporating the semirings from theoretical computer science.

Over [[near-rings]], one can consider near-ring modules, a nonabelian generalization of modules.{{Citation needed|date=May 2015}}