# zbMATH — the first resource for mathematics

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.
##### MSC:
 11R27 Units and factorization
Full Text:
##### References:
 [1] ARNOLD I.: Ideale in kommutativen Halbgruppen. Recueil Math. Moscow 36 (1929), 401-407. · JFM 55.0681.03 [2] BOREVICH Z. L.-SHAFARVICH I. R.: Number Theory. Academic Press, Inc, Orlando, 1966 [ · Zbl 0145.04902 [3] CLIFFORD A. H.: Arithemtic and ideal theory of abstract multiplication. Bull. Amer. Math. Soc. 15 (1934), 326-330. · Zbl 0009.14903 [4] CLIFFORD A. H.: Arithmetic and ideal theory of commutative semigroups. Ann. of Math. (2) 39 (1938), 594-610. · Zbl 0019.19401 [5] HERRLICH H.: Topologische Reflexionen und Coreflexionen. Lectures Notes in Math. 78, Springer-Verlag, Berlin-Heidelberg-New York, 1968. · Zbl 0182.25302 [6] HERRLICH H.-STRECKER G. E.: Category Theory. Allyn and Bacon, Boston, 1973. · Zbl 0265.18001 [7] KUMMER E. E.: Zur Theorie der komplexen Zahlen. Berlin. Monatsber. (1845), 87-96 (Abgedruckt in: J. Reine Angew. Math. 35 (1847)) [8] KUMMER E. E.: Über die Zerlegung der aus Wurzeln der Einheit gebildeten komplexen Zahlen in ihre Primfactoren. J. Reine Angew. Math. 35 (1847), 327-367 · ERAM 035.0993cj [9] MITCHELL B.: Theory of Categories. Academic Press, New York and London, 1965. · Zbl 0136.00604 [10] POSPÍŠIL B.: Remark on bicompact spaces. Ann. of Math. (2) 38 (1937), 845-846. · Zbl 0017.42901 [11] RIBENBOIM P.: The work of Kummer on Ferma\?s last theorem. Proc. Conf., Prog. Math. 26 (1982), pp. 1-29. [12] SKULA L.: Divisorentheorie einer Halbgruppe. Math. Z. 114 (1970), 113-120. · Zbl 0177.03202 [13] SKULA L.: Fortsetzung stetiger Homomorphismen von $$\delta$$-Halbgruppen. J. Reine Angew. Math. 261 (1973), 71-87. · Zbl 0266.20067 [14] SKULA L.: Maximale $$\delta$$- und $$\delta_1$$-Kategorien. J. Reine Angew. Math. 274/275 (1975), 287-298. · Zbl 0308.20048 [15] WALKER R. C.: The Stone-Čech Compactification. Springer-Verlag, Berlin-Heidelberg-New York, 1974. · Zbl 0292.54001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.