Math 6310 • Algebra

Let \(R\) be a commutative integral domain with identity. Recall that an \(R\)-module is torsion
free if for any \(r\in R\) and \(m\in M\), \(rm=0\) implies that either \(r=0\) or \(m=0\).

\begin{enumerate}
\item[(a)]
Suppose \(R\) is a principal ideal domain and \(M\) and \(N\) are two \(R\) modules. Prove that if
\(M\) and \(N\) are torsion free, so is \(M\otimes_R N\).
\item[(b)]
Give an example of an integral domain \(R\) and two torsion free \(R\)-modules \(M\) and \(N\) such
that \(M\otimes_R N\) is not torsion free.
\end{enumerate}