An approximation theorem for semigroups with divisor theory. (Ein Approximationssatz für Halbgruppen mit Divisorentheorie.) (German) Zbl 0742.20060

Halter-Koch, Franz

Result. Math. 19, No. 1-2, 74-82 (1991).

Only commutative cancellative semigroups with units are considered. Let \(H\) be a semigroup and \(F(P)\) be the free abelian semigroup generated by the set \(P\). Then a semigroup homomorphism \(\partial: H\to F(P)\) is called divisor homomorphism if for all \(\alpha,\beta\in H\) the fact \(\partial(\alpha)|\partial(\beta)\) (in \(F(P)\)) implies \(\alpha|\beta\) (in \(H\)). A divisor theory for \(H\) means a divisor homomorphism \(\partial:H\to F(P)\) such that for every \(p\in P\) there is a finite number of elements \(a_ 1,\dots,a_ n\in H\) for which \(p\) is the greatest common divisor of \(\{\partial(\alpha_ 1),\dots,\partial(\alpha_ n)\}\). The notion of Krull semigroup was introduced in connection with conditions under which the semigroup ring \(R[H]\) would be a Krull ring and is a direct generalization of the notion of Krull ring. This notion is equivalent to the notion of semigroup with divisor theory. The main result of the paper is the proof of the approximation theorem for Krull semigroups which gives a new proof of the approximation theorem for Krull rings. [See: L. G. Chouinard, Can. J. Math. 33, 1459-1468 (1981; Zbl 0453.13005); U. Krause, Proc. Am. Math. Soc. 105, 546-554 (1989; Zbl 0692.20058).].

Reviewer: G.I.Zhitomirskij (Saratov)

MSC:

20M14	Commutative semigroups
20M25	Semigroup rings, multiplicative semigroups of rings
20M15	Mappings of semigroups
13F07	Euclidean rings and generalizations

Keywords:

commutative cancellative semigroups; free abelian semigroup; semigroup homomorphism; divisor homomorphism; divisor theory; semigroup ring; approximation theorem; Krull semigroups; Krull rings

Citations:

Zbl 0453.13005; Zbl 0692.20058

References:

[1]	N. Bourbaki, Commutative Algebra, Addison-Wesley (1973).
[2]	L. G. Chouinard II, Krull semigroups and divisor class groups, Canad. J. Math. 33 (1981), 1459–1468. · Zbl 0453.13005 · doi:10.4153/CJM-1981-112-x
[3]	R. Gilmer, Commutative semigroup rings, The University of Chicago Press (1984). · Zbl 0566.20050
[4]	F. Halter-Koch, Halbgruppen mit Divisorentheorie, Expo. Math. 8 (1990), 27–66.
[5]	H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Teil Ia, Jber. d. DMV 36 (1927), 233–311; Nachdruck: Physica-Verlag, Würzburg (1965). · JFM 53.0143.01
[6]	U. Krause, On monoids of finite real character, Proc. Amer. Math. Soc. 105 (1989), 546–554. · Zbl 0692.20058 · doi:10.1090/S0002-9939-1989-0953009-9
[7]	S. Lang, Algebra, Addison-Wesley (1984).
[8]	L. Skula, Divisorentheorie einer Halbgruppe, Math. Z. 114 (1970), 113–120. · Zbl 0177.03202 · doi:10.1007/BF01110320
[9]	L. Skula, On c-semigroups, Acta Arith. 31 (1976), 247–257. · Zbl 0303.13014