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Jf Characterization of Kxt AM by Extensions
L Characterisation of TorsionTodules by Tor
2 other sections not shown
analogous arbitrary assume assumption basis of R/S called commutative diagram commutative with exact complex homomorphism consequence corollary to theorem define definition denote diagram is commutative dimension direct sum direct summand dual bases element epimorphism exact rows exists a homomorphism Extn fact free module Frobenius extension Frobenius homomorphism functors Hom(R identity map implies induction injective resolution integer integral domain left basis lemma let f monomorphism morphism multiplication Nakayama automorphism natural isomorphisms p-dim p-dim^A phism proj ective projective module projective resolution Proof Proposition prove R-direct R-exact R-homomorphism R-injective R-left modules R-module R-projective R-respectively R,S R-respectively R,S)-projective R-right module R-tensored bihomomorphism R,S)-injective Regard the diagram Remark resp respectively right basis right ideal rows are exact S)-infective short exact sequence similarly special choice splits submodule tensor product Theorem torsion uniquely determined zero divisors