Tag Archives: unique factorisation

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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Composition Series

Positive integers can be uniquely factored as a product of primes. Here, we would like to prove a counterpart for modules. Now there are two ways to “factor” a module M; a more liberal way takes a submodule N which gives us composition … Continue reading

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Topics in Commutative Rings: Unique Factorisation (3)

Example 1: The Gaussian Integers Z[i] Let’s pick the norm function N : Z[i]-{0} → N where N(a+bi) = (a+bi)(a–bi) = a2+b2. We know that N is a multiplicative function, i.e. N(r)N(s) = N(rs). Instead of checking this by brute force, we write N(x) = x·xc, where (a+bi)c = a-bi is the conjugate of a+bi. It’s easy to … Continue reading

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Topics in Commutative Rings: Unique Factorisation (2)

In the previous article, we imposed certain finiteness conditions on the ring (specifically a.c.c. on principal ideals: that every increasing sequence of principal ideals is eventually constant), then proved that unique factorisation holds if and only if all irreducible elements … Continue reading

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Topics in Commutative Rings: Unique Factorisation (1)

Unique Factorisation: Basics Throughout this post, let R be an integral domain; recall that this means R is a commutative ring such that whenever ab=0, either a=0 or b=0. The simplest example of an integral domain is Z, the ring of integers. What’s of interest to … Continue reading

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