Definition 9.7.1.
Let \(A\) be an integral domain, and let \(M\) be a left \(A\)-module. The torsion submodule of \(M\) is
\begin{equation*}
{\rm Tor}(M)=\left\{m\in M:a\cdot m=0,\;\text{for some}\,0\neq a\in A\,\text{that may depend on }m\right\}.
\end{equation*}
If \(M={\rm Tor}(M)\) then we say that \(M\) is a torsion \(A\)-module.
If \({\rm Tor}(M)=\{0\}\) then \(M\) is said to be a torsion-free \(A\)-module.
Consider an \(\Z\)-module \(M=\Z\times\Z/2\Z\text{.}\) We show that the rank of \(M\) is \(1\text{.}\) Suppose that \(r=(n,\overline{m})\) and \(s=(k,\overline{l})\) are two distinct elements of \(M\) with either \(n\neq 0\) or \(k\neq 0\text{.}\) We then have the following relation that implies that \(r,s\in M\) are \(\Z\)-linearly dependent.
\begin{equation*}
2k(n,\overline{m})-2n(k,\overline{l})=0\in M
\end{equation*}
If both \(n=0\) and \(k=0\) then
\begin{equation*}
2(0,\overline{m})+2(0,\overline{l})=0\in M
\end{equation*}
implies that \(r,s\) are \(\Z\)-linearly dependent. Thus the cardinality of a maximal linearly independent subset of \(M\text{,}\) i.e., \({\rm rank}(M)\leq 1\text{.}\)
Since \(\{(1,0)\}\) is \(\Z\)-linearly independent, we get that \({\rm rank}(M)=1\text{.}\)
However, \(M\) is not isomorphic to \(\Z\) as a \(\Z\)-mdoule. Indeed, \(M\) has nonzero torsion elements while \(\Z\) does not have any nonzero torsion elements.
Let \(V\) be a finite-dimensional vector space over a field \(F\text{,}\) and let \(T\in\End_F(V)\text{.}\) Using \(T\) we consider \(V\) as a left \(F[X]\)-module:
\begin{equation*}
F[X]\times V\to V\quad\text{given by}\quad \left(\sum a_iX^i,v\right)\mapsto\sum a_iX^i\cdot v=\sum a_iT^i(v).
\end{equation*}
By definition
\begin{equation*}
{\rm Tor}(M)=\left\{v\in V:\sum a_iT^i(v)=0\quad\text{for some }\sum a_iX^i\in F[X]\right\}.
\end{equation*}
By the Cayley-Hamilton’s, the characteristic polynomial of \(T\text{,}\) \(\chi_T\) annihilates \(T\text{,}\) i.e., \(\chi_T(T)(v)=0\) for every \(v\in V\text{.}\) Hence \({\rm Tor}(V)=V\text{.}\)
