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Let $R$ be an integral domain and $M$ an $R$-module. An element $m \in M$ is called a torsion element if $\operatorname{Ann}_R(m) \neq 0$, i.e. if $rm =0$ for some nonzero $r\in R$. Denote the set of torsion elements in $M$ by $T(M)$, called the torsion submodule of $M$. We say that $M$ is torsion-free if $T(M)=0$.

(i) Show that $T(M)$ is a submodule of $M$.

(ii) $M/T(M)$ is torsion-free.

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Have you tried anything for either part? –  Ben West Feb 19 '13 at 18:24
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Hi, welcome to MSE! Please try and use LaTeX formatting conventions for your posts. Also, please avoid formulating questions bluntly like this. Rather, explain what your thoughts are, what have you done so far, and where you are stuck. –  Andreas Caranti Feb 19 '13 at 18:27
    
I have deleted all the wrong tags. –  Martin Brandenburg Feb 19 '13 at 19:10
    
Why I have the feeling that you and Mathproof P. are one and the same user? –  user26857 Feb 19 '13 at 21:40

1 Answer 1

up vote 1 down vote accepted

Part (i) should be a straightforward verification.

For (ii), suppose $x+T(M)$ is a torsion element of $M/T(M)$. There exists nonzero $r\in R$ such that $r(x+T(M))=rx+T(M)=T(M)$. So $rx\in T(M)$. Thus there exists nonzero $s\in R$ such that $s(rx)=(sr)x=0$, so...

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