Module intuition

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pizzazz
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Module intuition

Postby pizzazz » Tue Apr 23, 2013 4:56 am UTC

So, in algebra we're covering modules over (commutative) rings, and I feel like I have no real intuition for them. They are defined in a way that generalizes vector spaces over fields, but then when we do stuff with them, we often do things that we do with rings (working with ideals and quotients and such) rather than what we do over vector spaces (linear transformations, for example). What is a module over a ring? What, intuitively, happens when you mod a module out by a submodule? I have a very general picture of this for rings, where certain elements become the "same" element, but I don't know what this would mean for a module (if it is at all different). Is there some good way to combine my intuition of rings and of vector spaces?

Thanks for your help in advance.

edit--we seem to be focusing in particular on modules over polynomial rings with coefficients in some field.

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z4lis
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Re: Module intuition

Postby z4lis » Tue Apr 23, 2013 12:42 pm UTC

Your intuition for rings carries over. We want to make all the elements of the submodule "the same", and we do this in a way that works well with the algebra, so the remaining object is still a module. Think about vector spaces, maybe R^3. If I pick a vector subspace V (which is precisely a submodule) and quotient by it, I'm identifying all vectors that differ by an element of that subspace. This ends up being something like projecting onto the orthogonal complement W of your subspace, since I can represent any vector uniquely by a sum v + w. For rings, modding out by an ideal is actually a special case of modding out by a submodule. It's a good exercise to work this out.

In some sense, modules generalize rings and vector spaces. Since ideals of rings are modules over the parent ring, modules also give you a nice language to talk about ideals, rings, and vector spaces all at the same time, which is nice.

You might also like this: you've probably seen group actions on sets. Think about how you might see a module as a ring action on an abelian group.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

DeGuerre
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Re: Module intuition

Postby DeGuerre » Wed Apr 24, 2013 12:12 am UTC

I like the analogy with a group action. If it helps, linear transformations on a vector space form a module, too. (Think about it: linear transformations form a ring, and vectors with addition form an abelian group.)

The notion of an "action" (e.g. group action, ring action, module) is probably one of the most important things you will ever understand in algebra, because it pervades maths and physics. (Warning: handwavy over-generalisation follows.) The world doesn't come equipped with a natural coordinate system, so the only measurements you can make are differences.

Rulers don't measure absolute position, they measure the distance between two positions. Clocks don't measure absolute time, they measure the duration between two times (and need to be manually set to a reference time, which depends on where you are in the world). Voltimeters don't measure absolute electrical potential, they measure potential difference. You get the idea.

In the real world, you need to impose a coordinate system to do anything useful. You need to define one time as "time zero", or one location as "the origin", or one direction as "north". Imposing a coordinate system is an "action".

(As an aside, because your coordinate system is artificial, any coordinate system imposed on the physical world should give you the same result no matter which one you chose. By imposing one coordinate system in particular, something has to give. This is where conservation laws, such as conservation of momentum, or conservation of energy, or conservation of electric charge, come from. But this is the maths forum, not the physics forum, so we won't go into this here.)

But here's what I really wanted to point: As you go on with abstract mathematics, you should prepare to lose hope of your intuition helping you out.

The reason why mathematicians do this abstract stuff: the same patterns keep turning up in extremely different theories of interest. So it makes sense to define the pattern formally so you can reason about it independently of any specific application.

A module, when you get down to it, is its formal definition, no more and no less. In a sense, you don't need intuition, you just need to work with it enough that it's a familiar concept on its own.

It only gets worse. Category theory abstracts all of abstract algebra, and lets you talk about (for example) the empty set, "false" in logic and the trivial ring as if they're all the same thing, the "initial object". And then it abstracts that in terms of an even more general concept, the "universal object". (That sounds as high-level as it's possible to get. You won't be surprised by now to learn that it's not.)

To understand that level of abstraction, you already need to be able to throw around the word "ring" as if it's a native concept without referring to analogies for it.

So my advice, which is worth exactly what you paid for it, is not to look for intuition. Instead, go through every example of a module you can think of, and work with it until the notion of a "module" becomes second nature.

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PM 2Ring
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Re: Module intuition

Postby PM 2Ring » Wed Apr 24, 2013 3:51 am UTC

DeGuerre wrote:The notion of an "action" (e.g. group action, ring action, module) is probably one of the most important things you will ever understand in algebra, because it pervades maths and physics. (Warning: handwavy over-generalisation follows.) The world doesn't come equipped with a natural coordinate system, so the only measurements you can make are differences.

Rulers don't measure absolute position, they measure the distance between two positions. Clocks don't measure absolute time, they measure the duration between two times (and need to be manually set to a reference time, which depends on where you are in the world). Voltimeters don't measure absolute electrical potential, they measure potential difference. You get the idea.

In the real world, you need to impose a coordinate system to do anything useful. You need to define one time as "time zero", or one location as "the origin", or one direction as "north". Imposing a coordinate system is an "action".

John Baez has a nice introductory article on this stuff: Torsors Made Easy.

DeGuerre
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Re: Module intuition

Postby DeGuerre » Wed Apr 24, 2013 5:45 am UTC

Yeah, I was going to drop the word "torsor", but then I looked at the Wikipedia page for it and thought it might not help.

BTW, John Baez is awesome.

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z4lis
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Re: Module intuition

Postby z4lis » Fri Apr 26, 2013 11:40 am UTC

DeGuerre wrote:So my advice, which is worth exactly what you paid for it, is not to look for intuition. Instead, go through every example of a module you can think of, and work with it until the notion of a "module" becomes second nature.


I've found this is sound advice for learning any mathematics. Think about the fuzzy intuition stuff for a while, and then write down examples. Prove each theorem. Apply it to the examples. It's amazing how much clearer a theorem becomes when you have a few stupid examples/counterexamples on hand. Do every exercise in the book. Then you'll understand modules better.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

pizzazz
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Re: Module intuition

Postby pizzazz » Sat Apr 27, 2013 9:49 pm UTC

Awesome, thanks for all the help guys!

Tirian
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Re: Module intuition

Postby Tirian » Sun Apr 28, 2013 3:25 am UTC

PM 2Ring wrote:John Baez has a nice introductory article on this stuff: Torsors Made Easy.


Wow. I have two books that introduce modules (Jacobson and Scott), but neither of them mentioned that affine groups are examples of modules despite going deep with examples. That helps a lot.


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