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Generalizations

A ring R corresponds to a preadditive category R with a single object. With this understanding, a left R -module is just a covariant additive functor from R to the 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) and consider the sheaves of O -modules (see sheaf of modules). These form a category O - 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 (X).

One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. 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.