Let R be a Dedekind domain whose quotient field K is an algebraic number field, and let Λ be an R-order in a semisimple K-algebra A with 1. A Λ-lattice is a finitely generated R-torsionfree left Λ-module. We shall call a Λ-lattice M locally free of rank n if for each maximal ideal p of R, Mp is Λp,-free on n generators. (The subscript p denotes localization.) The (locally free) class group of Λ is the additive group C(Λ) generated by symbols
where
and where xM = 0 if and only if M is stably free (that is, M + Λ(k) ≅ Λ + Λ(k) for some k).