Group rings over Dedekind rings. (English) Zbl 0657.13009
There are examples, due to J. Krempa [Proc. Am. Math. Soc. 83, 459–460 (1981; Zbl 0471.16007) and Can. J. Math. 34, 8–16 (1982; Zbl 0424.16005)] of non-isomorphic rings \(A\) and \(B\) such that the infinite cyclic group rings \(A[X,X^{-1}]\) and \(B[X,X^{-1}]\) are isomorphic. Here it is shown that if \(A\) and \(B\) are commutative Dedekind domains with \(A[X,X^{-1}]\cong B[X,X^{-1}]\) then \(A\cong B\). The proof, which is by contraction, uses an embedding of \(B\) in \((A/P)[X,X^{-1}]\), where \(P\) is a maximal ideal of \(A\), and field extension techniques which then become applicable.
Reviewer: D. A. Jordan
MSC:
13C05 | Structure, classification theorems for modules and ideals in commutative rings |
13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |
16S34 | Group rings |
References:
[1] | C. Chevalley,Introduction to the theory of algebraic functions in one variable, Math. Surveys, 6, AMS, 1951. · Zbl 0045.32301 |
[2] | Hartshorne, R., Algebraic Geometry (1977), Berlin: Springer-Verlag, Berlin · Zbl 0367.14001 |
[3] | Krempa, J., Isomorphic group rings with non isomorphic coefficient rings, Proc. Am. Math. Soc., 83, 459-460 (1981) · Zbl 0471.16007 · doi:10.2307/2044096 |
[4] | Krempa, J., Isomorphic group rings of free abelian groups, Canad. J. Math., 34, 8-16 (1982) · Zbl 0424.16005 |
[5] | Lang, S., Algebra (1977), Reading, Mass.: Addition-Wesley, Reading, Mass. |
[6] | Seghal, S. K., Topics in Group Rings (1978), New York: Marcel Dekker, New York · Zbl 0411.16004 |
[7] | Yoshida, On the coefficient ring of a torus extension, Osaka J. Math., 17, 769-782 (1980) · Zbl 0467.13015 |
[8] | Zariski, S., Commutative Algebraic Geometry, Vol. 1 (1958), Berlin: Springer-Verlag, Berlin · Zbl 0081.26501 |
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