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: 685 Thanks: 461 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  
Senior Member Joined: Aug 2012 Posts: 2,427 Thanks: 760  Quote:
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...tativealgebra 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 twodimensional 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  
Senior Member Joined: Sep 2016 From: USA Posts: 685 Thanks: 461 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
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 
Senior Member Joined: Aug 2012 Posts: 2,427 Thanks: 760  Did I? I just noted that it's hard to find pure examples. Today I learned! 
October 11th, 2018, 12:01 AM  #6 
Senior Member Joined: Oct 2009 Posts: 912 Thanks: 354  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 finitedimensional 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  
Senior Member Joined: Aug 2012 Posts: 2,427 Thanks: 760  Quote:
A pure example is a realworld application of modules that doesn't rely on the fact that vector spaces, which do have wellknown applications, are modules. Last edited by Maschke; October 11th, 2018 at 04:51 AM.  
October 11th, 2018, 06:25 AM  #8  
Senior Member Joined: Oct 2009 Posts: 912 Thanks: 354  Quote:
Aside, I think if you're in a math class that covers modules, you really shouldn't need motivation about the "realworld applications" anymore....  
October 11th, 2018, 10:45 AM  #9 
Senior Member Joined: Aug 2012 Posts: 2,427 Thanks: 760  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 
Senior Member Joined: Sep 2016 From: USA Posts: 685 Thanks: 461 Math Focus: Dynamical systems, analytic function theory, numerics  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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