A categorical contribution to the Kummer theory of ideal numbers

This article is partly a brief survey of known results which are going back as far as E. E. Kummer (1847), then to modern algebraic language of Z. I. Borevich and I. R. Shafarevich (1964) introducing the notion of theory of divisors, and to author's results using categorical methods in this area.The presented conception is chosen for better understanding the motivation of the new results and the notions.The main result of this paper is the description of all maximal S 1 -categories by means of so called a-ultrapseudofilters and ultrastars.A 5-^ -category is a subcategory M. of the category C of all S x -semigroups (which are semigroups possessing a divisor theory in the sense of Arnold) with semigroup homomorphisms, having the same objects as £, containing 8* -homomorphisms (defined by means of v -ideals) as morphisms, and with the divisor theory as a reflection for the reflective subcategory of M of all semigroups with unique factorization.It is shown that these maximal 5 X -categories form a set with cardinal number equal to expexpN 0 , while all the S 1 -categories form a class which is not a set.