Adèles and integral representations

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 $\Lambda$.