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Tag Archives: indecomposable modules
Idempotents and Decomposition
Let R be a general ring, not necessarily commutative. An element x∈R is said to be idempotent if x2 = x. Note An endomorphism f of an Rmodule M (i.e. ) is an idempotent if and only if f is a projection, i.e. M = ker(f) ⊕ im(f) and f … Continue reading
Posted in Notes
Tagged blocks, idempotents, indecomposable modules, modules, primitive idempotents
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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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Local Rings
Mathematicians are generally more familiar with the case of local commutative rings, so we’ll begin from there. Definition. A commutative ring R is said to be local if it has a unique maximal ideal. Note that every nonzero commutative ring has … Continue reading
Posted in Notes
Tagged algebra, associative algebra, indecomposable modules, local rings, units
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