An element \(m\) of the \(R\)-module \(M\) is called a torsion element if \(rm=0\) for some non-zero element \(r \in R\) .

(i) Prove that if \(R\) is an integral domain then \(T\) or \((M)\) , the set of torsion elements of \(M\) , is a submodule of \(M\) (called the torsion submodule of \(M\)).

(ii) Give an example of a ring \(R\) and an \(R\)-module \(M\) such that \(T\) or\((M)\) is not an \(R\)-submodule.

(iii) If \(R\) has zero-divisors, show that every non-zero \(R\)-module has non-zero torsion elements.

- Voclasses