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

Question

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.

Answer 1

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...