Let \(R\) be an integral domain and \(M\) an \(R\)-module. Define \[ T(M)=\{m\in M:rm=0\text{ for some }0\ne r\in R\}. \] Prove that \(T(M)\) is an \(R\)-submodule of \(M\).
Solution
1. One has \(0\in T(M)\).
For any nonzero \(r\in R\), \(r0=0\).
2. If \(m,n\in T(M)\) and \(c\in R\), then \(cm+n\in T(M)\).
Choose nonzero annihilators \(a,b\in R\) with \[ am=0, \qquad bn=0. \] Since \(R\) is a domain, \[ ab\ne0. \] Then \[ ab(cm+n) =bc(am)+a(bn) =0. \] Thus \(cm+n\) is torsion.
3. Therefore \(T(M)\le M\).
2. The domain hypothesis is used exactly to ensure that the product of two nonzero annihilators is still nonzero.