If $R$ is a integral domain then $TM$ is torsion module

Question

Let $R$ be an integral domain. An $R$-module $M$ is torsion if $TM = M$.

Prove that $TM$ is torsion.

I'm confused with this exercise, i need to prove that $TTM = TM$ but we know that $TM$ is a submodule if $R$ is an integral domain then $TM ≤ M $. Please some help for this, ty.

Answer 1

To show that $T(TM) \subseteq TM$, observe that

\begin{align*} m \in T(TM) & \iff m \in TM \text{ and there exists } r \in R : rm = 0 \\ & \implies m \in TM \end{align*}

and to show that $TM \subseteq T(TM)$, observe that

\begin{align*} m \in TM & \iff m \in M \text{ and there exists } s \in R : sm = 0 \\ & \implies m \in TM \text{ and there exists } s \in R : sm = 0 \\ & \iff m \in T(TM). \end{align*}

Conclude that $T(TM) = TM$.