Geodesic
Ad\`eles and integral representations
Izvestiya. Mathematics , Tome 3 (1969) no. 5, pp. 1019-1026.

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We apply the technique of adèles to study integral representations belonging to the same genus. We study the stable structure of genera and prove that if $L$ is a representation of a semisimple $Z$-ring such that each direct summand occurs at least twice in the decomposition of $L$ over the field of rational numbers, and if $M$ and $N$ are representations from the genus of $L$, then $M\oplus L^n\simeq N\oplus L^n$ implies that $M\simeq N$. For representations of a semisimple $Z$-ring $\Lambda$ we give a bound for the number of representations in a genus; the bound depends only on the rational algebra $\widetilde\Lambda=\Lambda\otimes Q$ and on the exponent of the group $\Lambda'/\lambda$ , where $\Lambda'$ is a maximal overring of $\Lambda$. 
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     title = {Ad\`eles and integral representations},
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Yu. A. Drozd. Ad\`eles and integral representations. Izvestiya. Mathematics , Tome 3 (1969) no. 5, pp. 1019-1026. http://geodesic.mathdoc.fr/item/IM2_1969_3_5_a4/

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