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On Central Ω-Krull Rings and their Class Groups

Published online by Cambridge University Press:  20 November 2018

E. Jespers  and
P. Wauters
E. Jespers
Affiliation:
Katholieke Universiteit Leuven, Leuven, Belgium
P. Wauters
Affiliation:
Katholieke Universiteit Leuven, Leuven, Belgium
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The aim of this note is to study the class group of a central Ω-Krull ring and to determine in some cases whether a twisted (semi) group ring is a central Ω-Krull ring. In [8] we defined an Ω-Krull ring as a generalization of a commutative Krull domain. In the commutative theory, the class group plays an important role. In the second and third section, we generalize some results to the noncommutative case, in particular the relation between the class group of a central Ω-Krull ring and the class group of a localization. Some results are obtained in case the ring is graded. Theorem 3.2 establishes the relation between the class group and the graded class group. In particular, in the P.I. case we obtain that the class group is equal to the graded class group. As a consequence of a result on direct limits of Ω-Krull rings, we are able to derive a necessary and sufficient condition in order that a polynomial ring R[(Xi)i∊I] (I may be infinite) is a central Ω-Krull ring.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 2 , 01 April 1984 , pp. 206 - 239
Copyright
Copyright © Canadian Mathematical Society 1984

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