Problem 3: An R-mod. M is called torsion free if M₂ = 0, where M₁ = {m EM | 3rЄR Arm = 0} Let R be an integral domain and M be an R-module. 1. Prove that M, is a submodule of M. 2. Give an example to show that part 1 is not true if we drop the domain condition. 3. Prove that free R-modules are torsion free. Is the converse true? 4. Prove that if M is divisible, then M, is also divisible. 5. If f Є HomR (M, N), then f(M) ≤ Nt. In other words; if f; denotes the restriction of f at M,, then fr Є HomR (Mt, Nt). 6. If 0 M' MM" is exact sequence of modules, then 0 →M' MM" is exact, where f, is the restriction off at M', and g, is the restection of g at Mr.