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

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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?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
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