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Tag Archives: leftexact
Projective Modules and Artinian Rings
Projective Modules Recall that Hom(M, ) is leftexact: for any module M and exact , we get an exact sequence Definition. A module M is projective if Hom(M, ) is exact, i.e. if for any surjective N→N”, the resulting HomR(M, N) → HomR(M, N”) is … Continue reading
Posted in Notes
Tagged artinian, free modules, leftexact, projective modules, semisimple rings, splitting lemma
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Tensor Product over Noncommutative Rings
Following the earlier article on tensor products of vector spaces, we will now look at tensor products of modules over a ring R, not necessarily commutative. It turns out we have to distinguish between left and right modules now. Indeed recall … Continue reading
Posted in Notes
Tagged bimodules, hom functor, leftexact, modules, rightexact, tensor products
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Hom Functor
Fret not if you’re unfamiliar with the term functor; it’s a concept in category theory we will use implicitly without delving into the specific definition. This topic is, unfortunately, a little on the dry side but it’s a necessary evil to get … Continue reading
Posted in Notes
Tagged bimodules, hom functor, left modules, leftexact, modules, right modules
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