Tag Archives: rings

Semisimple Rings and Modules

After discussing simple modules, the next best thing is to look at semisimple modules, which are just direct sums of simple modules. Here’s a summary of the results we’ll prove: A module is semisimple iff it is a sum of simple … Continue reading

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Simple Modules

We briefly talked about modules over a (possibly non-commutative) ring R. An important aspect of modules is that unlike vector spaces, modules are usually not free, i.e. they don’t have a basis. For example, take the Z-module given by Z/2Z. [ Recall: a Z-module is … Continue reading

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Elementary Module Theory (I)

Modules can be likened to “vector spaces for rings”. To be specific, we shall see later that a vector space is precisely a module over a field (or in some cases, a division ring). This set of notes assumes the … Continue reading

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Topology: More on Algebra and Topology

We’ve arrived at the domain where topology meets algebra. Thus we have to proceed carefully to ensure that the topology of our algebraic constructions are well-behaved. Let’s look at topological groups again. Our first task is to show that the … 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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