The Jacobson radical

The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even fails to be semisimple! A less restrictive notion, but one that still captures the notion of a ring which can be understood by how it acts on simple (left) modules, is that of a semiprimitive or Jacobson semisimple ring, one [...]

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Structures on hom-sets

Suppose I hand you a commutative ring . I stipulate that you are only allowed to work in the language of the category of commutative rings; you can only refer to objects and morphisms. (That means you can’t refer directly to elements of , and you also can’t refer directly to the multiplication or addition [...]

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Boolean rings, ultrafilters, and Stone’s representation theorem

Recently, I have begun to appreciate the use of ultrafilters to clean up proofs in certain areas of mathematics. I’d like to talk a little about how this works, but first I’d like to give a hefty amount of motivation for the definition of an ultrafilter. Terence Tao has already written a great introduction to [...]

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The adjoint functor theorem for posets

Recently in Measure Theory we needed the following lemma. Lemma: Let be non-constant, right-continuous and non-decreasing, and let . Define by . Then is left-continuous and non-decreasing. Moreover, for and , . If you’re categorically minded, this last condition should remind you of the definition of a pair of adjoint functors. In fact it is [...]

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The connected components functor

I skimmed through books 1, 4, and 5 of my new batch and am currently skimming through 3; it seems I don’t have the mathematical prerequisites to get much out of 2. It will take me a long time to digest all of the interesting things I’ve learned, but I thought I’d discuss an interesting [...]

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The ideal-variety correspondence

I guess I didn’t plan this very well! Instead of completing one series I ended one and am right in the middle of another. Well, I’d really like to continue this series, but seeing as how finals are coming up I probably won’t be able to maintain the one-a-day pace. So I’ll just stop tagging [...]

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The induced representation

Charles Siegel over at Rigorous Trivialities suggested a NaNoWriMo for math bloggers: instead of writing a 50,000-word novel, just write a blog post every day. I have to admit I rather like the idea, so we’ll see if I can keep it up. Continuing the previous post, what we want to do now is to [...]

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Some adjoint functors

Note: as usual, I will be playing free and loose with category theory in this post. Apologies to those who know better. One way to define a subgroup of a group is as the image of a homomorphism into . Given the inclusion map , the functor in the category of groups acts contravariantly to [...]

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