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The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f(m) for all elements m of M. [4] The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules.
Given left R-module E and right R-module F, there is a canonical homomorphism θ : F ⊗ R E → Hom R (E ∗, F) such that θ(f ⊗ e) is the map e′ ↦ f ⋅ e, e′ . [ 13 ] Both cases hold for general modules, and become isomorphisms if the modules E and F are restricted to being finitely generated projective modules (in particular free ...