On embedding of lattices belonging to the same genus
Abstract: If $R$ is an order in a semisimple algebra over a Dedekind ring and $M,\;N$ two $R$-lattices in the same genus, an upper bound for the length of the composition series of $M/N’$ for $N’ \cong N$, is given. This answers a question posed by Roĭter.
H. Jacobinski, Genera and decompositions of lattices over orders, Acta. Math. 121 (1968), 1-29.
A. V. Roĭter, Integer-valued representations belonging to one genus, Izv. Akad. Nauk SSSR Sen Mat. 30 (1966), 1315-1324; English transl., Amer. Math. Soc. Transl. (2) 71 (1968), 49-59. MR 35 #4255.
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Keywords: Representation of orders over a Dedekind ring, genus of representation modules, isomorphism classes in a genus, Dirichlet’s theorem on arithmetic progressions
Article copyright: © Copyright 1970 American Mathematical Society