Regularity in Categories

Part of the Frontiers in Mathematics book series (FM)


The definition of a regular map makes perfectly good sense in any category. To wit, let C be a category. We write AC 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 fC(A, B) is regular if and only if there exists gC(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 1AC(A) for any object A in the preadditive category C and C(A, M) is a C(M)-C(A)-bimodule.


Linear Transformation Division Ring Endomorphism Ring Additive Subgroup Nonzero Idempotent 
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© Birkhäuser Verlag AG 2009

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