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Stable torsion theories
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1-1 correspondence 1-topology abelian category ACC on right Ann(x assertions are equivalent class of modules cogenerated commutative diagram complemented Corollary define dense right ideals direct sum endomorphism ring essential right ideal essential submodule exact sequence exists F-closed modules F-injective F-submodule finitely generated right following assertions following properties Giraud subcategory Goldie torsion hence hereditary torsion theory idempotent implies indecomposable injective envelope injective module isomorphism Jacobson radical Ker f lattice left adjoint left exact preradical Lemma Let F Math maximal right modules of quotients monomorphism morphism nilpotent noetherian ring non-singular non-zero-divisor pretopology Proof Proposition QF-ring quotient rings regular ring right annihilators right finite-dimensional right noetherian right non-singular right self-injective ring homomorphism ring of quotients satisfies ACC self-injective ring semi-perfect semi-primary semi-simple subcategory of Mod-A subring suffices to show Suppose Theorem topology topology F torsion class torsion module torsion-free torsion-free modules two-sided ideal zero