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

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The discussion focuses on Exercise 8(c) from Section 10.1 of Dummit and Foote's "Abstract Algebra," specifically regarding torsion elements in modules. It establishes that for a non-zero R-module M, there exist non-zero elements a and b in R such that ab=0. The analysis divides M into two subsets based on whether bm=0 or not, concluding that if the first subset contains a non-zero element, M has non-zero torsion elements. The conversation encourages further exploration of the case where the first subset is empty. This highlights the importance of understanding torsion elements in module theory.
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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
 
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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 $\{m\in M: bm=0\}=\{0\}.$
 
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Thread 'How to define a vector field?'

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