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Krull Semigroups and Divisor Class Groups

Published online by Cambridge University Press:  20 November 2018

Leo G. Chouinard II
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R. Matsuda has shown that a group ring is a Krull domain if and only if the coefficient ring is a Krull domain and the group is a torsion-free abelian group satisfying the ascending chain condition (ace) on cyclic subgroups [6]. D. F. Anderson has used this to obtain a partial determination of when a semigroup ring is a Krull domain, and under certain circumstances to describe the divisor class group of such a ring ([1], [2]). Using some of Anderson's techniques, but taking a different approach, we arrive at a complete answer of a different nature to these questions. We call a semigroup satisfying the major new conditions arising a Krull semigroup, and define its divisor class group.

In particular, every abelian group is the divisor class group of such a ring, and it follows that every abelian group is the divisor class group of a quasi-local ring, which seems to be a new result.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 6 , 01 December 1981 , pp. 1459 - 1468
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Anderson, D. F., Graded Krull domains, Comm. in Alg. 7 (1979), 79–106.Google Scholar
2. Anderson, D. F., The divisor class group of a semigroup ring, Comm. in Alg. 8 (1980), 467–476.Google Scholar
3. Anderson, D. F. and Ohm, J., Valuations and semi-valuations of graded domains, preprint.Google Scholar
4. Bourbaki, N., Commutative algebra, (Addison-Wesley, Reading, Mass., 1972).Google Scholar
5. Claborn, L., Every abelian group is a class group, Pac. J. Math. 18 (1966), 219–222.Google Scholar
6. Matsuda, R., On algebraic properties of infinite group rings, Bull. Fac. Sci. Ibaraki Univ. 7 (1975), 29–37.Google Scholar
7. Samuel, P., Lecture on unique factorization domains, Tata Institute, Bombay (1964).Google Scholar