2 3 M ay 2 00 6 Relative Regular Objects in Categories

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the theory of generalized inverses of matrices is presented. Some of the techniques we use are new, since dealing with arbitrary categories allows us to pass to the dual category. Von Neumann regular rings play a fundamental role in Ring Theory, see [4]. A ring R is called von Neumann regular if for any a ∈ R there exists b ∈ R such that a = aba. This concept was generalized to modules in [7]. A left module M over the ring R is called regular if for any m ∈ M there exists g ∈ Hom R (M, R) such that g(m)m = m. Basic properties of regular modules are developed in [7]. Since a morphism f ∈ Hom R (R, M) is uniquely given by an element m ∈ M, one can reformulate the regular condition as follows. For any f ∈ Hom R (R, M) there exists g ∈ Hom R (M, R) such that f = f • g • f. This suggests the definition of a more general concept of an U-regular object in a category, where U is a given object of the category. We study basic properties of regular objects in categories, with special emphasis on abelian categories and on locally finitely generated Grothendieck categories. The main source of inspiration for our results was [7]. However, even for results that sound similarly, as for example our key Theorem 2.8, stating that a finite direct sum of U-regular objects in an abelian category is also U-regular, and extending [7, Theorem 2.8], the proof is consistently different. For some results we use Mitchell Theorem to reduce to categories of modules, but most of the proofs are done inside the general abelian category. We 1