Another Question on Torsion Elements .... D&F Section 10.1, Exercise 8 ....

Click For Summary
SUMMARY

The discussion focuses on Exercise 8(c) from Section 10.1 of "Abstract Algebra" (Third Edition) by David S. Dummit and Richard M. Foote, specifically regarding torsion elements in modules. The key conclusion is that in a non-zero R-module M, if there exist non-zero elements a and b in R such that ab=0, then M must contain non-zero torsion elements. The solution involves analyzing the subsets of M based on the action of b on elements of M.

PREREQUISITES
  • Understanding of R-modules and their properties
  • Familiarity with torsion elements in algebraic structures
  • Knowledge of the definitions and implications of zero divisors in rings
  • Basic proficiency in reading and solving exercises from abstract algebra textbooks
NEXT STEPS
  • Study the concept of torsion elements in detail within the context of module theory
  • Review the definitions and examples of zero divisors in ring theory
  • Explore additional exercises in Dummit and Foote's "Abstract Algebra" to reinforce understanding
  • Investigate the implications of non-zero R-modules in various algebraic contexts
USEFUL FOR

Students of abstract algebra, particularly those studying module theory, as well as educators and tutors looking for rigorous solutions to exercises in Dummit and Foote's textbook.

Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ...

I am currently studying Chapter 10: Introduction to Module Theory ... ...

I need some help with an Exercise 8(c) of Section 10.1 ...

Exercise 8 of Section 10.1 reads as follows:https://www.physicsforums.com/attachments/8313Can someone please demonstrate a rigorous solution to 8(c) ...

Peter
 
Physics news on Phys.org
Hi Peter,

We know from the assumption that there are non-zero $a,b\in R$, such that $ab=0.$ Now let $M$ be a non-zero $R$-module. Note that we have

$M=\{m\in M: bm=0\}\cup \{m\in M: bm\neq 0\}.$

Since $M$ is a non-zero $R$-module, one of these two sets must contain a non-zero element. If the first contains said non-zero element, then $M$, by definition, has non-zero torsion elements since $b\neq 0.$ I will leave it to you to work on the case where $\{m\in M: bm=0\}=\{0\}.$
 
Last edited:

Similar threads

MHB How Can One Show That a Given Example is Not a Submodule?
  • · Replies 4 ·
Replies
4
Views
2K
MHB (R/I)-Modules .... Dummit and Foote Example (5), Section 10.1 ....
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad (R/I)-Modules .... Dummit and Foote Example (5), Section 10.1
  • · Replies 7 ·
Replies
7
Views
2K
MHB Example on Z-modules .... Dummit & Foote, Page 339 ....
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Example on Z-modules .... Dummit & Foote, Page 339 ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Peter Needs Help on Cox et al - Section 8.1, Exercise 3(a)
  • · Replies 3 ·
Replies
3
Views
2K
MHB Help with Dummit & Foote Exercise 1, Section 13.2 - Algebraic Extensions
  • · Replies 2 ·
Replies
2
Views
2K
MHB Elementary Algebraic Geometry: Dummit & Foote Ch.15, Ex.24 Coordinate Ring
  • · Replies 1 ·
Replies
1
Views
2K
MHB Tensor Algebras - Cooperstein Example 10.1
  • · Replies 1 ·
Replies
1
Views
1K
MHB Projective Algebraic Geometry - Exercise 4(a) Cox et al - Section 8.1
  • · Replies 2 ·
Replies
2
Views
2K