**Category Theoretical Description**

Every ring can be thought of as a monoid in **Ab**, the category of abelian groups (thought of as a monoidal category under the tensor product of -modules). The monoid action of a ring *R* on an abelian group is simply an *R*-module. Essentially, an *R*-module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a "vector space over a ring".

Let (*A*, +) be an abelian group and let End(*A*) be its endomorphism ring (see above). Note that, essentially, End(*A*) is the set of all morphisms of *A*, where if *f* is in End(*A*), and *g* is in End(*A*), the following rules may be used to compute *f* + *g* and *f* **·** *g*:

- (
*f*+*g*)(*x*) =*f*(*x*) +*g*(*x*)

- (
*f***·***g*)(*x*) =*f*(*g*(*x*))

where + as in *f*(*x*) + *g*(*x*) is addition in *A*, and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring, (*R*, +, **·** ), (*R*, +) is an abelian group. Furthermore, for every *r* in *R*, right (or left) multiplication by *r* gives rise to a morphism of (*R*, +), by right (or left) distributivity. Let *A* = (*R*, +). Consider those endomorphisms of *A*, that "factor through" right (or left) multiplication of *R*. In other words, let End_{R}(*A*) be the set of all morphisms *m* of *A*, having the property that *m*(*r* **·** *x*) = *r* **·** *m*(*x*). It was seen that every *r* in *R* gives rise to a morphism of *A* - right multiplication by *r*. It is in fact true that this association of any element of *R*, to a morphism of *A*, as a function from *R* to End_{R}(*A*), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian X-group (by X-group, it is meant a group with *X* being its set of operators). In essence, the most general form of a ring, is the endomorphism group of some abelian X-group.

Read more about this topic: Ring (mathematics)

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