## The Morita theorems |

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A-A-bimodule A-A-isomorphic A-homomorphism A-submodules AŽKA Ae-endomorphism algebraic apply Morita Auslander and Goldman Aut(A bimodule Brauer group categories of left category theory central separable K-algebra class group classes of A-A-progenerators commutative ring convention Corollary cosets defines a bisection denote direct sum direct summand division ring elements endomorphism epimorphism extend by linearity finitely generated projective Galois group given by left given by right group of K-automorphisms HomA HomA(AP,AA HomA(P,P homomorphism Horn identity functor induces injective inner automorphisms isomorphism types K-Aut K-Aut(A K-categories K-functor K-isomorphisms lattice left A-module left isomorphism classes left multiplication Lemma map g measures the failure module Moreover Morita Context MORITA THEOREMS morphisms natural transformation non-zero number field opposite ring progenerator projective K-module Proof proper monomorphism Proposition Center right B-module right multiplication ring of nxn Rosehberg-Zelinski show that Ker sided ideals simple Artinian ring subgroup submodule sum of copies summand of F surjective symmetric zero