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 October 10th, 2018, 08:08 PM #1 Newbie   Joined: Oct 2018 From: Indonesia Posts: 1 Thanks: 0 Module theory in abstract algebra Hello guys, my lecture ask me to find the application of module theory in real life. Anybody can give me some example?
 October 10th, 2018, 08:50 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 503 Thanks: 281 Math Focus: Dynamical systems, analytic function theory, numerics Vector spaces are specific cases of modules and its hard to avoid applications of vector spaces in real life.
October 10th, 2018, 08:56 PM   #3
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 Originally Posted by SDK Vector spaces are specific cases of modules and its hard to avoid applications of vector spaces in real life.
A lot harder to think of a pure example. I know how to use a free module to construct the tensor product of two modules. That's literally the only application of modules I know. Tensor products are used in engineering and physics, but those are usually tensor products of finite-dimensional vector spaces. I have no idea what anyone uses modules for outside of abstract algebra. I think it's a pretty strange question from a math professor.

OP, is this for a math class? Or perhaps a computer science class? That could be a possibility.

ps -- I found this: https://mathoverflow.net/questions/6...tative-algebra

This is mathoverflow, a site for professional mathematicians. The question was what are the applications of commutative algebra, which is essentially the study of modules over a commutative ring. The only positive response was: The book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by Cox, Little & O'Shea, contains some "real world" applications, specifically chapter 6 (of the 3rd edition) is titled "Robotics and Automatic Geometric Theorem Proving"..

I think that's pretty abstract for the level this question was asked.

You'd have to really stretch. For example a grocer sells apples and oranges. Every day he sells an integer number of each (could be negative, maybe someone delivers more than he sells that day). So nA + mO is the expression for the number of apples and oranges he sells, and the set of all such expressions is a the two-dimensional free module over the integers. If I had to answer this question and couldn't fall back on vector spaces, this is the best I could do. Can anyone think of an application of $\mathbb Z$-modules that isn't overkill, such as the free module generated by apples and oranges?

Last edited by Maschke; October 10th, 2018 at 09:31 PM.

October 10th, 2018, 09:38 PM   #4
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Quote:
 Originally Posted by Maschke A lot harder to think of a pure example. I know how to use a free module to construct the tensor product of two modules. That's literally the only application of modules I know. Tensor products are used in engineering and physics, but those are usually tensor products of finite-dimensional vector spaces. I have no idea what anyone uses modules for outside of abstract algebra. I think it's a pretty strange question from a math professor. OP, is this for a math class? Or perhaps a computer science class? That could be a possibility. ps -- I found this: https://mathoverflow.net/questions/6...tative-algebra This is mathoverflow, a site for professional mathematicians. The question was what are the applications of commutative algebra, which is essentially the study of modules over a commutative ring. The only positive response was: The book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by Cox, Little & O'Shea, contains some "real world" applications, specifically chapter 6 (of the 3rd edition) is titled "Robotics and Automatic Geometric Theorem Proving".. I think that's pretty abstract for the level this question was asked. You'd have to really stretch. For example a grocer sells apples and oranges. Every day he sells an integer number of each (could be negative, maybe someone delivers more than he sells that day). So nA + mO is the expression for the number of apples and oranges he sells, and the set of all such expressions is a the two-dimensional free module over the integers. If I had to answer this question and couldn't fall back on vector spaces, this is the best I could do. Can anyone think of an application of $\mathbb Z$-modules that isn't overkill, such as the free module generated by apples and oranges?
I don't understand why you are disqualifying vector spaces? My point was that vector spaces are modules and have tons of applications so any one of them would answer the question.

There are also applications of modules which aren't vector spaces but as you have mentioned they are much more sparse. One example is computing homology for cubical complexes which is heavily used in topological data analysis.

October 10th, 2018, 09:54 PM   #5
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 Originally Posted by SDK I don't understand why you are disqualifying vector spaces?
Did I? I just noted that it's hard to find pure examples.

Quote:
 Originally Posted by SDK There are also applications of modules which aren't vector spaces but as you have mentioned they are much more sparse. One example is computing homology for cubical complexes which is heavily used in topological data analysis.
Today I learned!

October 11th, 2018, 12:01 AM   #6
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 Originally Posted by Maschke Did I? I just noted that it's hard to find pure examples.
What is a pure example?

Modules can be used to generalize some nice theorems and give nicer proofs of other theorems. For example, the classification theorem of finitely generated modules of a PID generalizes both
1) The classification of finite(ly generated) abelian groups
2) The classification of finite-dimensional vector spaces

and can further be used to give proofs of stuff like the rational and the Jordan canonical form, which can then further be used to solve systems of linear differential equations.

Sure, all of this can be done without modules. But modules offer a nice context to look at these different ideas combined. Same thing with modules in representation theory, they offer a nice fundamental context to look at some results. Sure, you can do without modules, but with modules is just nicer.

October 11th, 2018, 04:48 AM   #7
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 Originally Posted by Micrm@ss What is a pure example? Modules can be used to generalize some nice theorems and give nicer proofs of other theorems. For example, the classification theorem of finitely generated modules of a PID generalizes both 1) The classification of finite(ly generated) abelian groups 2) The classification of finite-dimensional vector spaces and can further be used to give proofs of stuff like the rational and the Jordan canonical form, which can then further be used to solve systems of linear differential equations. Sure, all of this can be done without modules. But modules offer a nice context to look at these different ideas combined. Same thing with modules in representation theory, they offer a nice fundamental context to look at some results. Sure, you can do without modules, but with modules is just nicer.
Real life? Only if you're in abstract algebra class. Read OP's question and consider its context. I surely never said modules weren't important in math.

A pure example is a real-world application of modules that doesn't rely on the fact that vector spaces, which do have well-known applications, are modules.

Last edited by Maschke; October 11th, 2018 at 04:51 AM.

October 11th, 2018, 06:25 AM   #8
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 Originally Posted by Maschke Real life? Only if you're in abstract algebra class. Read OP's question and consider its context. I surely never said modules weren't important in math. A pure example is a real-world application of modules that doesn't rely on the fact that vector spaces, which do have well-known applications, are modules.
Sure, but I think my example of the Jordan canonical form surely is important, not even in math class, but it has many very concrete applications. It's just difficult going into these applications without getting started on differential equations and the applications they can have.

Aside, I think if you're in a math class that covers modules, you really shouldn't need motivation about the "real-world applications" anymore....

October 11th, 2018, 10:45 AM   #9
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 Originally Posted by Micrm@ss Aside, I think if you're in a math class that covers modules, you really shouldn't need motivation about the "real-world applications" anymore....
I totally agree, which is why I said it was a strange question for a math professor to ask. Didn't I write that? Now I'm starting to wonder if I write one thing and people read something entirely different.

October 11th, 2018, 10:50 AM   #10
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 Originally Posted by Maschke I totally agree, which is why I said it was a strange question for a math professor to ask. Didn't I write that? Now I'm starting to wonder if I write one thing and people read something entirely different.
This may be true in characteristic 0 but I'm not sure why the characteristic matters as long as the Jacobian is invertible.

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