SEMIVARIETIES AND SUBFUNCTORS OF THE IDENTITY FUNCTOR

We study certain subcategories called semivarieties and obtain Kaplanasky's theorem on the decomposition of Abelian groups into a divisible group and a reduced group under the frame-work of category theory; We also investigate the con-nection of these epicoreflective subcategories with varieties. Semivarieties are subcategories of a category with cartain axioms; such subcategories play an important role in Abelian categories and a description of these by means of coreflection, appears in the general context in the work of Mitchell [8, §5, §6, III]. Broader classes than these have also been studied by Amitsur [l], Carreau [2], under the title of HI — RI properties of radicals and their classes. Our aim in this note is to give a categorical proof of Kaplanasky's Theorem 3 [6, §5] and while so doing we generalize the concepts of varieties and variety functors of Frohlich [4] under abstract frame work utiliz-ing Maranda's [9] concept of a radical. ^ is a category equipped with the following axioms: I. ^ has a null object. II. Every morphism a in ^ , admits a factorization a = vμ, where v is a normal epimorphism and μ is a monomorphism; we are writing the composition in the precise way • —>.—*. where the dots are the unnamed objects. ^f' III. Every family of objects has a direct and a free product. IV. The subobjects and factor objects of any objects form a set. It is immediate that & admits null morphisms. The image of a morphism in ^ , defined in axiom II and some time written as (v, L, μ) in the factorization.- ̂>L- ̂>, is uniquely determined to within equi-a valence. Every family of subobjects of an object i in ^ has a union and as such every morphism has a kernel. Dual consideration holds for factor objects and cokernel. A map admitting null subobject as the kernel is a monomorphism and a sequence 0>A^-*B-^C>0 is exact if and only if a is a monomorphism, β is a normal epimor-phism and the subobject (A, a) serves as the kernel of β. For details on the notation used and results mentioned in this section the reade