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Arithmetical properties of idempotents in group algebras. (English. Abridged French version) Zbl 0958.20007
In this nice paper the author studies for a finite group $$G$$ and a Dedekind ring $$A$$ of characteristic $$0$$ the possible denominators for the numbers $$e_g\in A$$ which can occur for an idempotent $$e=\sum_{g\in G}e_gg\in AG$$. In fact the author proves that given a prime ideal $$\wp$$ of $$A$$ and an idempotent $$\widetilde e$$ in $$A_\wp G$$, then there is an idempotent $$e\in A_\wp G$$ and a natural number $$n\in\mathbb{N}$$ so that $$A_\wp Ge=A_\wp G\widetilde e$$ and $$|G|^ne\in AG$$. To show the result one first semi-localizes outside the primes dividing the order of $$G$$ and different from $$\wp$$. Then, the observation is to show that an $$AG$$-lattice is indecomposable if and only if its localization at $$\wp$$ is indecomposable. From this one constructs the idempotent $$e$$.
MSC:
 20C10 Integral representations of finite groups 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16S34 Group rings
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