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A categorical contribution to the Kummer theory of ideal numbers. (English) Zbl 1062.11068
This article is partly a survey of known results which are going back as far as to E. E. Kummer (1847), and in modern language to Z. I. Borevich and I. R. Shafarevich (1964) introducing the notion of the theory of divisors, and to the author’s results (1973–1975) using categorical methods in this area. The main result of this paper is the description of all maximal \(\delta _1\)-categories by means of so called \(\alpha \)-ultrapseudofilters and ultrastars. A \(\delta _1\)-category is a subcategory \(\mathcal M\) of the category \(\mathcal L\) of all \(\delta _1\)-semigroups (which are semigroups possessing a divisor theory in the sense of Arnold) with semigroup homomorphisms, having the same objects as \(\mathcal L\), containing \(\delta ^*\)-homomorphisms (defined by means of \(\nu \)-ideals) as morphisms, and with the divisor theory as a reflection for the reflective subcategory of \(\mathcal M\) of all semigroups with unique factorization. It is shown that these maximal \(\delta _1\)-categories form a set with cardinal number equal to exp \(\aleph _0\), while all the \(\delta _1\)-categories form a class which is not a set.
11R27 Units and factorization
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