Regularity and Substructures of Hom pp 131-156 | Cite as

# Regularity in Categories

## Abstract

The definition of a regular map makes perfectly good sense in any category. To wit, let *C* be a category. We write *A* ∈ *C* if *A* is an object of *C* and by *C*(*A, B*) we denote the set of morphisms in *C* for objects *A* and *B* in *C*. Then *f* ∈ *C*(*A, B*) is **regular** if and only if there exists *g* ∈ *C*(*B,A*) such that *f g f = f.* To do more we need more special categories. A **preadditive category** is a category *C* in which the morphism sets are abelian groups with the property that composition of morphisms distributes over addition. It is also postulated that the category contain a null object 0 such that *C* (0, *B*) = {0} and *C*(*A*, 0) = 0. This assures that *C*(*A*) := *C*(*A, A*) is a ring with 1_{A} ∈ *C*(*A*) for any object *A* in the preadditive category *C* and *C*(*A, M*) is a *C*(*M*)-*C*(*A*)-bimodule.

## Keywords

Linear Transformation Division Ring Endomorphism Ring Additive Subgroup Nonzero Idempotent## Preview

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