Prevarieties in Abelian categories

Abstract

Following [1] we define a radical in an Abelian category ~ to be a subfunctor of the identity functor p: R ~id ~ such that RR(A) =R(A) and R(A/R(A)) = 0 for every object A of ~. A radical R is called a torsion if for every object B and each of its subobjects A there holds the equation R(A) :A NR(B), to be understood in the sense that the square R (B) -+ R (A) Bu---~A is Cartesian. An object A in an Abellan category ~[ with radical R is called semisimple if R(A) =0. The complete subcategory of all semisimple objects in~ is called semisimple. As is well known, it is a prevariety [21. This paper gives an axiomatic description of the following classes of categories: the class ~p of categories that are equivalent to prevarieties of locally small Abelian categories, the class ~r (~t, respectively} of categories that are equivalent to semisimple subcategories of locally small Abelian categories with radical (with torsion, respectively}. On the basis of this description we characterize the classes ~, .~:, ~[ consisting of the categories that are equivalent to subcategories of the corresponding type in categories of right unitary modules over associative rings with unit.