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

[Math Amateur]

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

 

[GJA]

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