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Tag Archives: splitting lemma
Commutative Algebra 27
Free Modules All modules are over a fixed ring A. We already mentioned finite free modules earlier. Here we will consider general free modules. Definition. Let be any set. The free Amodule on I is a direct sum of copies of … Continue reading
Posted in Advanced Algebra
Tagged exact functors, free modules, leftexact, localization, projective modules, splitting lemma
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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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KrullSchmidt Theorem
Here, we will prove that the process of decomposing is unique, given that M is noetherian and artinian. Again, R is a ring, possibly noncommutative. Definition. A decomposition of an Rmodule M is an expression for nonzero modules An Rmodule M is said … Continue reading
Posted in Notes
Tagged indecomposable modules, krullschmidt, local rings, matrix rings, splitting lemma, unique factorisation
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