An approximation theorem for semigroups with divisor theory (Q1174613)

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: \textit{L. G. Chouinard}, Can. J. Math. 33, 1459-1468 (1981; Zbl 0453.13005); \textit{U. Krause}, Proc. Am. Math. Soc. 105, 546-554 (1989; Zbl 0692.20058).].