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 R-module 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

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Krull-Schmidt 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 non-commutative. Definition. A decomposition of an R-module M is an expression for non-zero modules An R-module M is said … Continue reading

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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 non-zero commutative ring has … Continue reading

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