My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Thanks Tree1Thanks
Reply
 
LinkBack Thread Tools Display Modes
October 10th, 2018, 07:08 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?
Jessica Rahma Prillantika is offline  
 
October 10th, 2018, 07:50 PM   #2
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 473
Thanks: 262

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.
SDK is offline  
October 10th, 2018, 07:56 PM   #3
Senior Member
 
Joined: Aug 2012

Posts: 2,044
Thanks: 584

Quote:
Originally Posted by SDK View Post
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 08:31 PM.
Maschke is offline  
October 10th, 2018, 08:38 PM   #4
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 473
Thanks: 262

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
Originally Posted by Maschke View Post
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.
SDK is offline  
October 10th, 2018, 08:54 PM   #5
Senior Member
 
Joined: Aug 2012

Posts: 2,044
Thanks: 584

Quote:
Originally Posted by SDK View Post
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 View Post
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!
Maschke is offline  
October 10th, 2018, 11:01 PM   #6
Senior Member
 
Joined: Oct 2009

Posts: 555
Thanks: 179

Quote:
Originally Posted by Maschke View Post
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.
Micrm@ss is offline  
October 11th, 2018, 03:48 AM   #7
Senior Member
 
Joined: Aug 2012

Posts: 2,044
Thanks: 584

Quote:
Originally Posted by Micrm@ss View Post
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 03:51 AM.
Maschke is offline  
October 11th, 2018, 05:25 AM   #8
Senior Member
 
Joined: Oct 2009

Posts: 555
Thanks: 179

Quote:
Originally Posted by Maschke View Post
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....
Micrm@ss is offline  
October 11th, 2018, 09:45 AM   #9
Senior Member
 
Joined: Aug 2012

Posts: 2,044
Thanks: 584

Quote:
Originally Posted by Micrm@ss View Post
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.
Maschke is offline  
October 11th, 2018, 09:50 AM   #10
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 473
Thanks: 262

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
Originally Posted by Maschke View Post
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.
Thanks from Maschke
SDK is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
abstract, algebra, module, theory



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
abstract algebra q1 mohammad2232 Abstract Algebra 1 January 25th, 2015 06:37 AM
Abstract Algebra micle Algebra 1 June 17th, 2013 09:06 AM
Abstract Algebra Need Help ustus Abstract Algebra 4 October 14th, 2012 12:00 PM
Abstract Algebra Help MastersMath12 Abstract Algebra 1 September 24th, 2012 11:07 PM
Abstract Algebra micle Abstract Algebra 0 December 31st, 1969 04:00 PM





Copyright © 2018 My Math Forum. All rights reserved.