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:
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| [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 |
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