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Regularity in Categories

Part of the Frontiers in Mathematics book series (FM)

Abstract

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.

Keywords

Linear Transformation Division Ring Endomorphism Ring Additive Subgroup Nonzero Idempotent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2009

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