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abelian group algebraic number field annihilator assume Cartan map central simple clearly commutative ring composition series Corollary cyclic Dedekind ring defined Definition diagram direct summand division ring epimorphism exact sequence finite group finite number finitely generated projective finitely generated torsion free A modules Frobenius functor Frobenius modules GL(A global field Gq(A Gq(R group of order Hence homomorphism idempotent implies integral closure isomorphism classes isomorphism mod torsion Jordan-Zassenhaus Theorem K-algebra kernel KQ(A Kq(R Let f matrix maximal ideal maximal order monomorphism morphism Nakayama's lemma noetherian number of possible phism prime ideal projective R module proof of Theorem Proposition prove quotient field R-module R-order ramified relatively prime Rff module ring with quotient root Seiten semisimple separable set of primes sided ideal simple A module simple algebra splits submodule Theorem 8.l0 theorem holds Theory