Finitely generated modules over a PID, and applications on abelian groups

In summary, the conversation was about a problem involving a submodule generated by elements satisfying certain relations in a free module. To understand this submodule, it was suggested to consider the matrix formed by the relations in Smith normal form and use the structure theorem to determine the torsion coefficients. However, the conclusion was not clear. It was suggested that the submodule is free and full, and the Smith form can help determine which basis vectors are in the kernel.
  • #1
Jösus
16
0
Hello!

I'm currently taking a course in group- and ring theory, and we are now dealing with a chapter on finitely generated modules over PIDs. I have stumbled across some problems that I can't really get my head around. It is one in particular that I would very much like to understand, and I would greatly appreciate some help.

The situation is the following. Assume we are given a free module, for example the [tex]\mathbb{Z}[/tex]-module [tex]\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}[/tex], and want to consider the submodule generated by all elements [tex]\left(x_{1},x_{2},x_{3}\right)[/tex] satisfying certain relations [tex]\sum_{1}^{3}{a_{i,j}x_{j}} = 0[/tex] for [tex]i=1,2[/tex]. I have heard that a good way of understanding this submodule would be to consider the matrix [tex](a_{i,j})_{2,3}[/tex], put it in smith normal form and then conclude that the entries on the generalized diagonal will be the torsion coefficients in the decomposition of the submodule as a direct sum of cyclic ones (as in the structure theorem). However, I have not managed to draw that conclusion. If someone could explain why this holds true, or if it is false tell me that (and perhaps give me a hint on how to understand said submodule), I would be extremely thankful.

Thanks in advance!
 
Physics news on Phys.org
  • #2
I am not sure about this but this is my thought.

The kernel of the linear map,

[tex]
(a_{i,j})_{2,3}
[/tex]

is a full free submodule of the original free module,M, and thus is generated by a linearlly independent set of vectors that can be extended to a basis for M.
The Smith form will tell you which basis vectors are in the kernel - after a suitable change of basis.

- Why is the submodule free? - because any submodule of a free module over a PID is free

- why is the submodule full? because if A.(rV) = 0 for any r in the PID the A.V = 0 by linearity.

does this seem right?
 

1. What is a PID?

A PID, or principal ideal domain, is a type of commutative ring in abstract algebra that has certain properties, including the existence of a unique factorization of elements into irreducible elements. Examples of PIDs include the integers, polynomial rings over a field, and certain rings of algebraic integers.

2. What is a finitely generated module?

A finitely generated module over a PID is an abelian group that can be generated by a finite set of elements. In other words, every element in the module can be expressed as a linear combination of a fixed number of basis elements. This is analogous to a vector space over a field, where every vector can be expressed as a linear combination of a finite set of basis vectors.

3. Why are finitely generated modules over a PID significant?

Finitely generated modules over a PID have many useful properties that make them important in various areas of mathematics, including abstract algebra, number theory, and algebraic geometry. They have a well-defined notion of rank, and they can be decomposed into a direct sum of cyclic modules. Additionally, they have applications in the study of abelian groups and the classification of finitely generated modules over other types of rings.

4. How are finitely generated modules over a PID related to abelian groups?

Every finitely generated abelian group is isomorphic to a direct sum of cyclic groups, and every finitely generated module over a PID can be decomposed into a direct sum of cyclic modules. This means that the study of finitely generated modules over a PID can be applied to the study of abelian groups, and vice versa.

5. What are some applications of finitely generated modules over a PID?

One major application is in the study of abelian groups, where finitely generated modules over a PID are used to understand the structure and classification of abelian groups. They also have applications in algebraic geometry, where they are used to study algebraic curves and surfaces. Additionally, they are used in coding theory, specifically in the construction of error-correcting codes.

Similar threads

A On a finitely generated submodule of a direct sum of modules....
    • Linear and Abstract Algebra
Replies
7
Views
1K
MHB Finite Sum of Indecomposable Modules .... Bland, Proposition 4.2.10 .... ....
    • Linear and Abstract Algebra
Replies
1
Views
895
Free Modules: Bland Corollary 2.2.4 - Issue on Finite Generation
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB Free Modules: Solving Issue of Finite Generation Corollary 2.2.4
    • Linear and Abstract Algebra
Replies
4
Views
2K
MHB Understanding Bland's Example: Free Modules & Directly Finite/Infinite R-Modules
    • Linear and Abstract Algebra
Replies
7
Views
2K
I Directly Finite and Directly Infinite R-Modules - Bland S2.2
    • Linear and Abstract Algebra
Replies
4
Views
2K
I Free Abelian Groups .... Aluffi Proposition 5.6
    • Linear and Abstract Algebra
2
Replies
40
Views
3K
MHB Modules - Decomposibiity of abelian groups
    • Linear and Abstract Algebra
Replies
4
Views
2K
Groups of homomorphisms of abelian groups
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB Free Abelian Groups .... Aluffi Proposition 5.6
    • Linear and Abstract Algebra
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top